It's very difficult to train a cat, especially one who is accustomed to
being an outside cat, to accept a harness or a collar. If you start with
them as a kitten, it is a lot easier. -  Good luck.......

I'm also thinking of getting my cat a harness. After a couple of weeks
in a pet hotel he has become brave and got used to walking free in the
open (the hotel has an open-air fenced walking ground). Back at home, he
has developed a habit of running down the stairs whenever the front door
opens, and the stairs end 50 centimetres from the road... Every time he
does it I get closer to a heart failure :(
I know that teaching him to walk like a doggie would be a hell of a job,
so wish me good luck.

Fitting the harness properly is crucial for a cat who has no clavicles
and could back through a needle if you gave her a chance.  The neck
part should be a strict two-fingers looser than the neck, no less and
no more.  The trunk part should be the same.  Any tighter, and
breathing becomes limited, but any looser, and they can immediately
back out of it.  Also the size of the harness you buy is important as
it determines the length of the center strap that connects the two
main loops.  The sizing is usually similar to their weight, but you
have to experiment depending on the blubber content of your cat, if he
has one.

You will have earlier success if you do not initially leave the cat
unsupervised with the harness on, but use it to go out with her on
walks, or sniffs, or whatever.  When he is more comfortable to having
it one without feeling constrained in where he can go, he will with
time be more biddable to being stuck on a long leash for a small
period of time.  Cats are not very tolerant to restraining, in
general, so you must be patient.  Lots of love, rewards and treats
after successfully putting the harness on will be a good start.

Good luck.  And sorry about your neighbour.

--tension