Think twice before you shoot a cat. I'm not weak in the gut about killing whatever needs killing, but there may be serious repercussions if you do it.
First problem, it's likely a felony. My neighbor shot a dog with a BB gun to scare it out of his yard. Kids saw him do it, told a teacher at school, CT State Police arrested him. The dog's owner told the police he would not press charges and felt that the neighbor was in the right. Teaching the dog a lesson without causing real harm. Police reply, "It's not up to you. The State will press charges." I don't know what the law is where you live, but you may want to check into it. Shooting a non game, domesticated animal is often classified as 'cruelty to animals' and in todays society you may be treated like a real criminal.
Next problem, cats don't die easy. I've shot rabbits and had them keel over on the spot. I hear moose dies easier that whitetail. I've killed whitetail but not moose, so that's hearsay. Cats take a lot of killing. A friend of mine had trouble with a ferrel tom, shot it with a 22. The cat ran, luckily cornering itself to a spot where he double tapped it. Flopping, yowling, more shots. He ended up emptying the High Standard into it. Moral of the story, cats often take more than one shot. That fact may lead to you getting caught. That brings us right back to problem one.
As I said before, I'm not squeamish. I own about 100 birds (my wife's hobby actually but I have to protect them) Exotic game birds are expensive and all kinds of animals want to eat them. I have to do what I have to do, even if it's someones beloved fluffy killing my stock. I have been lucky on that count and mostly shoot fox, and rats upon rats with the occasional other bird killer in the mix. I understand that you may need to kill a cat. Society and the law may not get it.
My take on air guns is go big. The .22 kills better than the .177 in my experience. Others may disagree, but that has been my experience. A cat may well run off anyway.
Consider a live trap for a nuisance animal that you don't want to get in trouble for. You could take him to the pound, or drown him in the bath tub where no one else will know.
There is an old saying among animal raisers about problem critters. "Shoot, shovel, and shut up."
Good luck. Stay out of trouble.
One last note, a felony 'cruelty to animals' charge would preclude you from EVER legally owning guns. Is this problem cat worth that? Buy or rent a live trap, dig a hole, and DON'T post the outcome on the internet.
