If I was in arena 4 but got demoted back to arena 3, will I still get the arena 4 cards?
While what agent86 said is true, if you have any chests over from the higher arena, they will still contain those arena's cards. So yes, you will still get arena 4 cards but you will not get arena 4 chests. Therefore, the only way to get the arena 4 cards would be if there were arena 4 cards in the shop before you were demoted, or if you had any arena 4 chests.
In fact, the answer is yes (to your title, at least), but only in a very specific circumstance. As I mentioned here - the Arena you play in determines what the awards will be for either player if they are the victor. You can therefore get a Chest from a higher Arena if you are high enough and get lucky (both by getting matched to a higher Arena, and winning that match).