My recommendation is to either choose a scope with adjustable objective, or a scope specifically designed for .22 rimfire. (Those are generally adjusted to be parallax free at 50 yards instead of the 100 or 150 parallax setting common to scopes designed for centerfire rifles). I really like my Nikon Prostaff 3-9x rimfire scope, but it is a little over your budget. I have a Simmons 3-9x32 rimfire scope with adjustable objective, but its optics are no match for the Nikon. Still for $60 or so, the Simmons does work. I actually prefer the Simmons 4x fixed rimfire scope for about $40 for a cheap scope. The fixed power tends to be more clear optically, and really for shots out to 60 or 70 yards, it has enough magnification. You might also take a look at the Center Point scopes offered at Walmart. I have read some pretty positive comments about those, and they are definitely within your budget.
NRA Endowment Member