First of all, shooting rats is little more than cheap entertainment. You will not curb their population that way, poison and/or trapping is the only thing to seriously drop their numbers especially if you have a farm nearby, you are likely dealing with a healthy rat population. Also they can get wise to shooting, I have opened the window and shot one under the bird feeder before and after you do it once or twice the others start to get smart and run when they hear or see the window crank open. I would use a bucket trap, fill bucket just under half full with water and sprinkle feed so it floats on top, the rats will fall in and can't get out eventually drowning. You can cover the bucket with chicken wire to keep chickens out but rats can go through the wire. There are examples of these traps on youtube.
As far as the bb/pellet gun, I would suggest a Crossman 2100 Classsic type pump up model or anything better than that like the stronger break action or pump up Benjamin Sheridan models. You will not want a Daisy Red Ryder type bb gun, they don't shoot very accurately and aren't too strong. Another option to consider since you have a .22 is to look around for some CB caps, they are a short 22 shell they don't have much gunpowder, only a primer and small charge, make no sound other than the click of your firing pin hitting the shell casing. They are around the strength of a good pellet gun and I believe more accurate as they are coming out of your rifled 22 barrel rather than a cheap smoothbore bb/pellet rifle. Get the 22 cb caps or cb shorts, not the cb longs, That would be your cheapest route.
Here is a link to she shells I am talking about, only 710 feet per second: http://www.midwayusa.com/product/1301376838/cci-ammunition-22-cb-short-29-grain-lead-round-nose
These shells aren't strong enough to cycle a semi auto 22 but can still be fired through them or any other 22lr rimfire by manually loading them into the chamber.
Edited by blucoondawg - 3/15/13 at 4:02pm