If you're on a budget, stay with iron sights. Spring piston air guns kill cheap scopes.
The Sheridan silver streak is a bit out of your budget, but a quality powerful air rifle that will do what you ask, but 20 cal. If you can find a used one or a benjamin pump 22, then get it. I'm still shooting my dad's benjamin air rifle that is ~80 years old. It did get a rebuild ~30 years ago, but is still going strong.
