You are newly sober, therefore you are in a very sensitive, anxious stage where everything seems worse than it is. If you are planning on remaining sober and not continuing to be loud and disruptive and they have never actually complained, I see no reason to say anything. It's never going to happen again, so the problem is gone. If they've not complained, then it's probably not an issue. Even if they had complained, I certainly don't suggest telling them you're an alcoholic. The reason behind noise from neighbours is irrelevant as long as it stops.
As for any one else, it really depends on the extent that you've hurt them? When you're a bit more clear headed you'll know if there are people you need to apologize to.
If you never drink again, none of these things can happen again.