First of all, my sympathies for the difficulties you are having to cope with. Bad patches do come along, but they usually don't last for ever. It just feels like it at the time....
The walls must be very thin where you live if you can hear all his little remarks, and if he can hear what you are saying inside your own flat. You say you think it's the guy below, but you don't sound sure. You need to establish that. If it is him, I'm not sure It would be a good idea to confront him. Us folks with AS find it hard to get our point over at the best of times, and you don't want a situation that's just awkward at present turning into something worse.
You are clearly finding all this problematic and distressing, but try and keep it in perspective. If all the guy is doing is moaning about you, that's really just a piece of trivia that doesn't affect you unless you let it. He's just sounding off and won't be dwelling on it, so why should you? Perhaps it has got bigger in your mind than it objectively is. If nobody is doing anything to you, threatening you or abusing you, there isn't actually a major problem, just an annoyance.
I would ask myself whether I am doing something that anyone else could reasonably consider annoying, eg. do I make a lot of noise? I only have your side of the story, but from what you say it sounds as if the guy below is the one with the problem, not you. If he lives alone, who is he talking to? And if he really is not a nice person, don't let him see that this is upsetting you or he might think he has control of the situation and start to bully you.
I don't see why it should make any difference to anyone else whether or not you go out much. If the man downstairs hardly ever goes out, he is no different to you in that respect. If you are not interfering with these people's lives, why should they have any problem with you?
I suggest being correct, pleasant and polite whilst holding him at a distance. I certainly wouldn't try to do things differently in order to please him, beyond of course behaving with normal consideration for others. I don't suggest actively trying to make friends either. If he has a problem with you, he should come to you and try to resolve it, but instead all he does is gripe about it. Why should you put yourself out to try and sort out a problem he seems to have but hasn't even told you about?
I hope this helps a bit. It seems inadequate but I'll post it anyway. Best of luck with getting things onto a smooth track!!