1.) Ask the neighbor to take care of his cat...in a nice way.
2.) A non-positive response? Then head over to the nearest Farm Store.
3.) Whip out wallet.
4.) Buy a properly-sized "Hav-A-Hart" trap. Keeps the caught critter alive.
5.) Bait and wait.
6.) Take trapped "rodent" to nearest Humane Society, turn over and that's that.
7.) Neighbor has to "pay" to get kitty back.
8.) A few trips and eventually, problem is solved, cat gets to waste its other 8 lives with owner...in IT'S house.