Well a bolt action rifle would be useful. Specially if they are Romero zombies. You have the advantage of range and accuracy. Almost useless at close range but the zombies shamble so as long as I can keep my self mobile, I shouldn't have a problem. I'd use one of the lower calibre rifles. Those that can be suppressed to the point of being pretty silent. No need for using a 50cal Anti Material Rifle on zombies, too heavy, too loud.
I find it a little stupid that you wouldn't take a side arm with you. Even if you have to travel light and you need the ammo space. Walking around with out at least a side arm for close encounters is a daft idea. I'm sure you can spare the room for a small handgun on your person. Also a crowbar would be useful, for getting passed locks and taking out lone zombies without making too much noise. Again, you can easily spare the ammo space for a crowbar (which is much more important than ammo given it's flexibility as both a silent weapon and a tool).
Actually if you was smart, you'd carry a back pack with you so you could carry extra ammo, food and drink, along with any tool that could be useful. Even if your travelling light... You'd be better off carrying a backpack full of supplies and as much ammo as you could find. Yes you may get tired a little faster, just take regular breaks, the ammo, food and tools are much more important.