I like your raise pre-flop. With only one late limper and the blinds left ATs is a good hand. The raise was probably too large though. 80 would have been better.

Top pair best kicker is a great flop and calls for a three quarter to full pot raise which you did. Now the surprise comes when he calls - what has he got? Pocket 5s and 6s are unlikely because he shouldn't have called your pre-flop raise. If he did then I'm prepared to lose all my chips. If he's got pocket 10s or above I'm also prepared to lose all my chips. That's poker.

Its even more unlikely that he called with 87, but not impossible.

Most likely holdings are K10, Q10, J10 pocket 9s and maybe 10-9.

The turn is where you made your mistake by not betting. You have to bet here to get information. If he's drawing to an open-ended straight you can't give him a free card by checking. If he's got pocket 9s a bet would likely make him muck. If he calls or (check raises) a big bet again then goes all in on the river you know you're beat and can throw your hand away.

Now he goes all in. when the nine flops. Your weakness on the turn makes it possible he is making a move. The fact he went all-in also points to it being a move. If the nine completes his nut straight or makes trips why go all in against someone who showed weakness on the turn. Same with two pair. I'd try extract money.

I still think he may have K10, Q10 or at worst maybe a Jack.
I'd call all in.