You listed the Ruger American .22WMR Rimfire.
If you really want to stick with a rimfire round, that's a good option for your stated price range.
However, if this is truly a hunting rifle, you can choose from other, more effective, calibers:
.30-06
.308 Win.
270 Win.
6.5 Creedmoor
.243 Win.
7mm-08 Rem.
Given the reloading options for centerfire ammo, I'd opt for one of those calibers.
Ruger American is at the top of most lists for light weight, budget-friendly, bolt action hunting rifles. Once you decide on this platform, the caliber is really the only important consideration left to decide.
If the size of the game you're likely to encounter is pretty small (birds), then a shotgun makes more sense than a rifle. Small game animals aren't always considerate enough to stop moving so you can make an easy rifle shot.