You will shoot your eye out.
IIRC, Crossman made pretty good pellet rifles back in the day, they’re still around:
It's a lot of gun for the money, and if you discover that .177 is not doing the job you can switch to .22.
I have a ‘Rat problem too, but it’s less amenable to an airgun solution.
I know you asked for air rifle information, but I also had many tree pests that were invading my 6 pecan trees. I set out 2 live catch traps and trapped over 20 the first year.
I don’t think I would of had the patience, nor the time, to have shot them instead.
I just seen one in Sportsman Guide, a Walther single cock .177 pellet, 1400 fps with a scope for $153 if you are a member, $170 if you are not.
Go with something in .22 cal and a break barrel action.
If cost isn't an issue, but accuracy is: http://www.pyramydair.com/s/m/AirForce_Ultimate_Condor_PCP_Air_Rifle/1469
Only because we live in a rural setting, and have all sorts of different situations than urban dwellers do I say “FWIW” here’s how I handle rats in trees: Rats in trees mean to me big nests consisting of trash, and twigs. I blast them out of the trees with a 3000 psi high pressure washer. Takes about 8-10 minutes of blasting from different angles to get the nests out of the trees, AND DOES NO HARM WHATSOVER TO THE BARK OF THE TREES AS SUGGESTED BY SOMEONE ON THE GARDENING THREAD A COUPLE OF YEARS AGO.
No nests, no rats. Last time I blasted the little b-stards nests was several years ago, and they never returned. I had three nests in three different trees around the property.
So if applicable to your situation OK, if not.....well good luck. I would believe the air guns would be only worth it if the rats make themselves available as targets often enough that you wouldn’t be wasting your time away from FREE REPUBLIC for hours waiting for them to appear so you could blast away at their worthless little carcasses.
I recommend any break-barrel, preferably with gas piston instead of spring. CO2 is illegal in most “city” areas, but break-barrel, within your property is okay. Airgun depot is good source. Reconditioned are good deals too.
I took a problem raccoon down with a 1200ft/sec Gamo, Squirrels, ie tree rats, are easy kills, and my dog loves them for a late afternoon snack.
If you live in town, you really shouldn’t be shooting a pellet gun into the air; if you live in the middle of no where, use a 410 or 20 gage shotgun.
First thing to remember is these claims of velocity are almost always highly inflated. They are made using extremely light alloy pellets and even then are probably off by a good bit.
The cheapest one I know of which is both accurate and powerful enough to get the job done is the Daisy 880 pumper. They are surprisingly accurate and around 800fps if you pump them 12 times. It sells fro $45 at Wal-Mart
For around $100 you can get the Crosman Quest. I have one and it is very accurate tho I am not sure if I just got a good one or not. They are made in China and I am told they are a copy of the Dianna 34. They are probably pushing 1000fps with lead pellets. It also comes with a scope.
I have another which would be ideal for your purposes if you don’t mind the weight and cost. It is a Dianna 48. It is both accurate and powerful, probably close to 1200fps with normal heavy pellets. It is made in Germany and it’s only downside other than cost is it is heavy.
Go to pyramidair.com, and find the best 22 calibeer spring type you can afford and buy it, along with some lead 22 caliber pellets.
.22 CB Caps in a .22 rifle with a cheap scope. They make almost no noise. You can put out an eye with them, so be careful.
My preference is larger .22 caliber and I like the multiple pump type gun, so I can choose the power for the situation.
The popular guns from my childhood are still around, so I went with a variable pump, bolt action Crossman/Benjamin 392. It's been around forever and parts and a few accessories are likely plentiful. Pumped the maximum six pumps, it's quite powerful, even more so than the one I had as a kid.
Like so many good things, one will lead to another, so...
I also bought a Crosman Nitro Venom Dusk Breakbarrel, gas piston single-shot. This, like other brands/models of this type, is a very powerful gun that shoots pellets, definitely not a toy.
Both of the above models are also available in .177 cal, which has higher muzzle velocities than the .22 cal.
Fun to shoot, even in the house on a rainy day (safety first, with care and a good backstop), with a wide variety of cheap and plentiful ammo, and powerful enough to handle small pests and game. (Bonus: the 392 will drive things off when pumped just once or twice, if you don't want to be lethal)