Out to 500 yards screams .243Win or other 6mm to me. The .22s are going to suffer a bit that far out.
You can't have a gun that's perfect for long shots and close snap shots at the same time, there are trade-offs to idealize for either scenario. If you like ARs that's one way to go, if you like bolt guns you can get a Savage and that will be really easy on your budget.
The easiest way to make sure you can do both close and far shots is to be careful on the optics you select. The Bushnell 6500 line offers a 2.5-16X magnification range, which would make for easy close shots and good magnification for 500 yds.