This one is my favorite. Known by many as a "mini sniper" rifle. The Mossberg Chuckster 640KD:
I've seen some very accurate and nice Marlin 882's, owned a very accurate RAR. It actually shot so well that a couple buddies ended up buying the Ruger RAR models. The Savage 93's are very good shooting and functioning rifles.
I actually saw a very nice 22wmr yesterday, that I'd like to buy. It was a now discontinued fancy maple stocked Browning T bolt 22WMR. It was priced at $700.00, so falls within the OP's required budget. That one felt great in the hands, and probably shoots lights out.
Everyone has listed some of their favorites, but one thing about it, it's not very hard to find a nice 22WMR for under $1,000.00..