Hammerman, if the cat is getting through the window, perhaps move the bin/ anything that it is climbing on to get in. Perhaps only have your top windows open if the problem is really bad, or screen off the bottom windows with net screens (buy of ebay) just a s you would to keep out the fyls. There is nothing your neighbour can do other than neutering the cat (presumably if they are not responsible to have already had the cat neutered, any approach you make would fall on deaf ears - not a lot other than neutering they can do to help the issue though unfortunately - if the cat is getting into your property it is your responsibility to ensure it can't get in).
Loud claps will scare the cat out of the house, but ideally, net screens would stop the issue altogether.
Shooting or trapping the animal is NOT an option, you are the more intelligent animal in this scenario - out-smart it :o)