OP didn't specify typical range, but when you said inside city limits, I'm not picturing a 5+ acre plot of land. Can you give more details on the typical range you see them at, OP?
Certainly be sure you do check your local laws to verify the legality of firing an airgun within city limits. I'm not so sure that there are "many places you can't even shoot a bb gun anymore," or what that claim might be based on - I just know its illegal to discharge a firearm, yet perfectly legal to fire an airgun, within the city limits where
I live.
You might also want to call to the city's animal control dept. If you're inside the city limits, they may have some suggestions, or even offer some sort of pest control service.
Regarding air rifles, if you do move ahead and buy one, I've had an outstanding experience with a Benjamin air rifle. This was about fifteen years ago - my last airgun gifted to me by my parents when I was preparing to enter my teens - but I've only heard great things about Benjamin since that time.
This (link) is the Benjamin Marauder - a .22 cal, 1000 FPS, PCP air rifle. It's priced here at $470. It has fifteen 5-star reviews, and none with anything less. To be completely transparent, you'd need a pump so you can fill it home, unless you happen to be into SCUBA and have a tank around that can get you ~2500 PSI. A pump will run you another $170 or so (
here's one offered at the same site), and you'd need an optic... Obviously this would put you over your budget, but if worst comes to worst and you've still got this problem, maybe this is an option you'd want to consider. And maybe, like others have said, it would be totally inadequate. At least now you have an good idea what you can get in the airgun world for $500.
This rifle might also be an option. Cheaper than the Marauder, at $300, and it's a break-barrel design, so no extra investment to shoot, as with the PCP rifle. Comes with a cruddy scope and no irons, but you could buy a pretty decent optic for it and still come in right at your $500 budget. 1100 FPS from this .22 cal rifle.
You might also consider archery. A bit of practice and an inexpensive recurve bow could eliminate your fox problem with perfectly humane kills and the acquisition of an inexpensive new hobby. You could
easily come in under your $500 budget. Heck, you could cut it in half and make do if you had to.