Depending on what Oppie considers “good distance away.”
If you consider say, 50yds or so, then a 12ga would do the trick.
Hell, you could use an AR15 for anything up to 400yds (effectively), but it’d cut a bit far into the $700 budget including ammo.
I stick w/ Mosin as the best option. You can get a 440rd spam can of it for $100. Only issues is the ammo is corrosive, so you’ll be spending more time cleaning. The ammo is also only FMJ for the most part, which isn’t great for attempting to kill something. Ideally you’d want something that expands upon contact. But if you are good enough to only get headshots, FMJ isn’t an issue.
Though I do wonder Oppie… All this zombie talk… You don’t happen to work for Umbrella Corporation, do you?