What arena will be my chest be, if I promote?

In my experience, the chest awarded from a win is based on the Arena played in, not on your personal arena. If you are a "Visitor" to another arena (which happens most often when you're near the next arena and play one of the lowest ranked opponents in that arena instead of your own) and win, you should receive that arena's chest.

Your Free and Crown Chests are based on your current Arena, though, and the situation where you can visit another Arena is very short-lived if you're actually winning Chests from it, because you'll promote to that Arena in a win or two.

The Clash Royale official website answers this question:

Chests won from battle are based on the Arena you fought in

So, it does not matter if you got promoted or not, or even what arena you are in. The only factor that decides which chest you get is what arena you fought in for that match.

The awarded chest will be of the arena that you are promoted to. So if you enter a higher arena, you are sure to have at least one chest from that arena.

(Source, personal experience. I went from arena 4 to 5, I was awarded an arena 5 Giant chest, luckily)