I don't think it's a tall order, as long you understand that most airguns are not all that quiet. If you want quiet so the neighbor won't hear I'd use Aguila Colibri 22LR ammo, or buy a much more expensive airgun. All the airguns in your price range, or even double that price will be heard by your neighbor.
If you accept the noise I think this one in the below link would be an excellent first gun, and on sale for $140. Add pellets to hit the free ship mark and you're good to go.
http://www.airgundepot.com/refurbished-benjamin-trail-22-caliber-nitro-piston-air-rifle-combo.html
As far as airguns in this price range it's as quiet as you can get. Another issue with cheap guns is accuracy will be a gamble, but that is usually fixable to a point with some work. If you want guaranteed accuracy out of the box you have to spend a whole lot more $.
This gun on sale, which is not on sale very often btw, is in your price range, quiet, and powerful.
The stock on this model is 16-24oz heavier than typical so the gun with scope is ~ 9lbs. The stock is softish plastic and very strong, so its soft semi-rubbery makeup is great for taming the shot cycle of the gun, helping accuracy and even quiets it a bit further. The drawback is the extra weight you have to drag around...
But like most of these Crosman/Benjamin guns they are the same base model so you can buy parts from the same base model if you want, and for cheap. Say you want to shave 24oz off with the wood Trail stock or one of many other stocks they sell. I prefer the Benjamin Regal stock which is only $23. You can also buy a cheaper gun like a Crosman Vantage (currently on sale for $86), then buy the Trail barrel (which is shrouded/suppressed) and you have a gun that is basically a Trail with a lighter wood stock but less $. It also comes with a more basic scope but that's up to you. Just know most Crosman/Benj guns are modular and parts are very cheap. Other brands are not so modular and parts are either expensive or impossible to get so basically you buy them as-is and that's it. If you're into tinkering to make the gun better then there's also a bunch of stuff you can do with Cros/Benj.
The NP2 gspn mentioned is a different base model with few interchangeable parts, like just a few different stocks which are expensive. The NP2 is a little more powerful but a lot more $. The NP2 has a nicer trigger but the Trail trigger can be modified to be the better one. The NP2 is supposed to be better quality but not by much and the end result to the end user is probably about the same.
So imo a lot depends on if you're willing to do some work to the gun, because you can also take an even cheaper gun like a $70 TR77 and make it even quieter than the Trail for about $35 and some work.
