You know, the two things I think about whenever I'm in a new place are (1) where's the best possible place to be in case of nuclear attack and (2) what's my zombie survival plan? So I've thought over this a great deal.
Frankly, I'm most likely screwed, depending on what kind of zombies they are. If they're the slow moving type, I've got a better chance. I live on the top floor so obviously the hardest part will be getting out of the building (my apartment isn't particularly secure enough--weak door and too many windows that open onto the roof). Now, I'm assuming the likeliest scenario is that the zombies will be coming up the stairwell, so obviously I'll want to avoid that. I'll move down the building by dropping from balcony to balcony until I'm at the bottom floor (or until I've slipped and fallen and killed myself) and then make my way out the building courtyard to the nearby river, which I'll follow until I come to the northern edge of town (my nearest exit point from the city). I'll start making my way north until I hit the Georgian border, at which point I'll tell the border guards what's going down since they're the nearest folks with weapons.
Now, if it's the fast moving variety, things become significantly more difficult. Obviously the stairwell is out, but likely so is the balcony option, since those fuckers can climb. My best strategy in this case will be to hide until they've come and looked through the apartment. I've got this perfect space in my couch that I can hide in and that no zombie would look. It's not under the couch, but is in fact a space for storage between the cushions and the bottom of the couch, and once tucked away a zombie would not see me even if it looked under the couch. Also, the cushion is heavy and takes some knowledge of how to work it to get it to open up, so they'd probably not get it. Once they left I would obviously wait for several hours to be sure that they were gone and wait until the light started to go; that way I'd be able to peek out the windows with minimal risk of detection. Then, the next day (bad idea to travel at night with fast moving zombies) after ensuring that the zombies have fairly well abandoned my section I'd take the same route as above. Unfortunately, it's likely they'd have reached the border by that time and I'd be unable to get to the guns; I'd try anyway, and if I found it impossible I'd likely find the nearest overrun base, stock myself up on weapons, find a secluded village that either hadn't been overrun or was already abandoned by the zombies, and hole up until it's likely that they'd starved to death.
Unfortunately, at this point I'm missing a primary tool: a machete. I'm not a big fan of guns in the zombie apocalypse, nor chainsaws, because you either run out of bullets or you run out of gas. Really, what you want is an incredibly sharp machete or sword, something that will go clean through the head or neck without getting stuck. I, unfortunately, still don't have one but I should really invest in one; otherwise all I've got is a tiny little axe that would almost surely get stuck in the first body it was chopped into.