Adnan,
You will likely get lots of well meaning advice on this subject. Here's mine, and it's based on some expensive, hard-won experience. CALL A REAL ESTATE ATTORNEY RIGHT NOW and have them review your case (voicemail, lease, damage estimate from 1st floor neighbor, etc.). There are too many variables to try and deal with this issue in a discussion forum. And frankly, while well-meaning, most replies from non-attorneys are likely to be based on emotion, heresay, or anecdotal past experience and not the law.
I happen to like Milt Colegrove for this. He is a Real Estate attorney and sponsor for our club. I've found his advice to be timely (he gets back with me very quickly), affordable (not the most important factor in choosing an attorney), accurate (in my opinion the most important factor when choosing an attorney), and based in "real world" Real Estate, and not just in the Law.
In my opinion, there are some practical matters you are going to have to deal with. Let's assume you "win" and get a judgement against your current tenant. What happens to you (and your threatened lawsuit from your 1st floor neighbor) when your tenant moves out and is unable or unwilling to pay?
From a practical perspective I suspect you are going to want to limit your monetary expenses by dealing with this as quickly as you can, and, assuming the attorney tells you that you are going to be the responsible party, by collecting what cash you can from your tenant and then solving the problem as quickly and cost effectively as possible with your downstairs neighbor.
Finally, I suggest you keep two things in mind. First, give the neighbor money, unless your attorney tells you otherwise, I would NOT fix his apartment. If you do, it's my understanding that if any of those repairs ever fail in the future, you will be held responsible for fixing them again. And second, any protracted court battle is likely to cost you more than rehabbing an apartment.
As the guy who started this club used to say, "Think with your pocketbook, and not your ego."
Let me know if there is anything else I can do to help you grow your business.
-Greg
Greg Wilson
The REI Mentor