Bushnell TRS-25 Trophy Series Red Dot Sight.
Well within your price range.
I use a Vortex Sparc red dot on mine, basically the same idea only a bit pricier. The TRS-25 are very well thought of/clear optics, and you can mount it very close to the bore so you're not fighting the arc of the shot much at 50yds.
On a basic 10-22 Carbine, the accuracy of a 0x Red Dot will exceed the accuracy of the rifle, in most cases, at 50yds. Unlimited eye relief and 0 parralax if you don't get yourself perfectly lined up, mean that it's far easier to get quick snap-shots off than with an actual scope. And the aim point with a Red Dot is far easier on the eyes than open sights.
Personally, I just don't find the 10-22 to be an accurate enough platform for actual magnified optics. 50yds minute of grouse head can be done with a red dot.