Jeep, water purifier, first aid kit, katana.
Wish I could take flashlight, but it's a 5th option on the list.
No reason to take guns since rounds will be gone quickly and you'll end up with a worthless piece of metal. Chainsaw is clunky and fuel-dependent. Fire axe is also clunky and actually not that good at chopping wood - not to mention you'll make a lot of sounds while doing so. Machete is bad on both fronts. Crossbow - almost perfect, but bolts tend to break too.
Jeep adds to mobility, which is very handy. Fuel-dependent, but then again, with such mobility you can reach some gasoline. Doubles as a place to stay in extreme situations, without losing mobility, rendering it able to escape emergencies.
Water purifier will allow you to move on without acute diarrhea
First aid kit is a must in literally every survival situation. You should be able to prevent infections and restore mobility as quickly as possible
Katana is too difficult to maintain, crossbow bolts can be fashioned by hand from found materials, first aid kit can be scavenged pretty easily. If this is starting gear, go for the stuff that is either rare or likely to be scavenged first, and that is easy for you to maintain by hand. Jeeps aren't rare, real body armor is. The machete is the go-to blade for many parts of the world for a reason. It is heavy enough to cleave small brush and limbs alike, sturdy enough that it isn't likely to break on you, and you can get a utility or survival version that has a saw on the back and other useful tools included in it. Only take the Jeep if you are proficient in maintaining it and know how to make biodiesel. If you have those skills, go for it. Your assessment is correct. If not, it is prioritizing short-term benefits over even medium-term. Even on a full tank, you have less than a day's travel in it. Then it becomes a hard-walled tent or tiny home, take your pick of feel-good terminology. Depending on your starting location, that half-day of travel may not be enough to get you safely away from population centers. If you are in any major US East Coast city, you are fucked. You can barely make it into the Appalachians on one tank, and certainly not remote enough to be safe. As you move further and further west, that becomes less and less of an issue. If you are in the SW, a Jeep becomes top tier pick because it lets you navigate offroad to somewhere truly remote and provides adequate hauling capacity for scavenged materials for setting up a secure base of operations.
Where are you, I'd love to analyze a different country?. Who knows, pigs may fly and I may be able to afford to be traveling abroad when Z-Day happens.
Looking at this population density map of Russia and the satellite map on Google, I feel like the Jeep would only be a viable take if you are somewhere in the medium-density band like the area around Neya. Looking further North, and correct me if I'm wrong, the terrain looks like it takes on a generally rough aspect. Looks like a decent amount of steep hills and mountains which lack infrastructure, so while the jeep could offroad it, the time it would take to traverse would be onerous and diminish the already abysmal fuel efficiency.
Exploring this forestry map I am seeing a lot of dense tall forests as you move north through the western part of the country, which is also looking like pretty rough terrain, so I am thinking the list I would go for in that region would be the fire axe, wrist rocket, winterized body armor, and climbing gear or a backpack. Clean water should not be as much of an issue given the number of lakes and accompanying tributaries in the region, plentiful wood supply for small fires for purifying water, and I'm pretty sure a lot of it is coniferous (correct?) which would mean that there should be ample tinder in dry needles so most fire kits would be overkill.
The real difficult decision is in the climbing gear, as being able fo ascend a cliff or tall tree and safely secure yourself would be ideal for Z-poc survival. Just ascend the tree, drive a piton in and fashion a sleeping harness from the climbing rope. Make sure you have a good amount of ammo for the wrist rocket and in the morning you can obliterate any Zs that have gathered around the bottom of the tree. Long term, I would totally go Ewok/Forest Elf with it and build a canopy encampment. Lots of space and easy-to-control access points, and with each additional platform it becomes easier and safer to build the next. The real issue with that is the ability to safely fell trees and hewn lumber, which would be nearly impossible solo unless you could adequately secure a perimeter so you didn't have to focus on threat vigilance. Barring that, I would say a clifftop camp with a rapid descent escape route. Good sight lines, effective egress, and if it is tall enough, you could try to get them to chase you and have an effective way to clear the Zs without wasting ammo. You would just need to clear the corpses after using it.
The reason why the question is tough with the pack is because of the terrain traversal, you need to be able to appropriately distribute the weight if anything you carry into wilderness like that, and that is tough without a good pack, especially with the body armor. I contemplated swapping the armor for a pack, but I can't justify being able to avoid bites and other injuries for traversal ease. If I could have 5, those would be the 5.
One thing though; the area to the north is very swamped; it's not just lakes. So getting dry lumber any time of the year is near impossible, thereby fire kit is very advisable. Dry needles are not as dry as one would want. Cones that are often advised as an element of easy fire starters are extremely wet and useless for the task; they burn no better than wood thorougly soaked in water. Generally, making fire out of any lumber in the woods is a massive headache in there.
Also, high humidity means fog is a very regular occurrence, so a clifftop might be way less useful that one can imagine. It gets particularly bad the closer you move to Khibiny mountains.
That definitely changes things. Swamps would likely be the last place you would want to try to survive out the Z-poc. The mobility issues would be useful for impeding zombies and raiders, but you would suffer the same issues until you could get some sort of infrastructure in. And that is completely ignoring the fact that Zs can end up hiding under the muck and bog. Also, the drinking water would be a real issue, since, as you said, everything is wet. If you could locate a good pete bog you could harvest and dry it in a shelter for amazing fuel, but I don't know how prevalent those are in those particular swamps. Other than resource gathering though, I think it would be best to avoid until you had a foothold somewhere else.
Now the fog in the mountains is actually interesting. That level of humidity means that you can source drinking water directly from the air with little issue. Cut up some cotton shirts so they are broad and flat, then rig up a wind vein that will keep a small sail aligned with the prevailing winds so the fog flows over it and the cloth will be soaked with fresh clean water. If the vein is built right you can even channel drip water into a container, then wring out the cloth when the fog has lifted before the sun can evaporate it. Variations on this are actually viable in any humid region, especially if the temperature dips below the dew point at night.
I think I would still probably make for the foggy mountains if I were in that region if I could do so while bypassing swampy regions. The rough terrain could be trivialized with minimal infrastructure and resources could be plentiful if managed right. I am also of the minor assumption that in that region the cold is almost as much of a threat as the zombies, so I'm not positive what sorts of shelter the mountains would provide. If memory serves, most of that region is very old metamorphic rocks that aren't prone to cave formation. This would severely limit shelter from the elements without a good source of building materials or the means to produce them. I would amend the list to fire axe, body armor (winterized), fire kit, wrist rocket. In scavenging, I would prioritize a survival machete, a pack, the aforementioned claiming gear, and real medical supplies (suture needle, sterile gauze, strong thread or monofilament fishing line). Once I am able to secure somewhere to put a cache, that expands to a charged car battery or 3, jumper cables, basic carpenter tools, many many tarps, and as many fluid containers as I can get my hands on. Russian winters are no joke and without the tarps, drying enough wood to survive would be hell. The tarps would he enough to make a sealed drying environment until a kiln dryer could be constructed. Of course, all of this is predicated on being able to secure an area that has materials and is remote enough that the possibility of wandering Zs becomes slim. It would probably he worth finding a camp in one of the transitional zones between the swamps and mountains so the swamp can act as a natural trap/barrier to Z traversal so it only requires minimal fortification.
I'd say there's enough stuff that can be turned into fire starter should you have time to dry it (and under a good cover with relatively strong winds and relatively low humidity for the area it will take several days to dry thin starters and months to dry firewood, so tarp won't help you that much in the short run). Generally, if you're in the wild in winter in northwestern Russia without already stored dry firewood or other fuel, your chances of survival are fairly questionable to begin with.
Yes, there aren't many caves in there and terrain is very hard. Constructing a cover is not a trivial task. Normally in this environment survivalists resort to self-made tents or dig into the ground (latter being less popular in the winter, for obvious reasons), but both are fairly useless in a zombie apocalypse for the reasons you've already outlined. Mountain and treetop constructions will suffer severely from strong cold winds.
Realistically though, you cannot survive the wilderness of northwestern Russia (and a lot of wilderness for that matter) with just 4 objects. You'll strike a way better chance by finding a remote village. Yes, you might have to ward off zombies, but, as you mentioned, environment will be less of an issue, and environment here is certainly deadly.
Yeah, that is a really rough place to survive in a non-apocalyptic scenario. I think focused active scavenging on the way out of civilization would be the key to making it. Then finding/setting up a village would be the next key to long-term survival.
I'm glad that I was able to reevaluate with better accuracy. This was fun.