PREFLOP: A9s is a borderline hand UTG, but I believe there are many party tables where limping here is profitable

FLOP: Youre getting 9-1 to call, I would estimate that you have 5 pure outs, which means you have the odds to call, but becuz there is a chance you may have the best hand here, raising is the best option. The BB could easily have AQ,AJ,AT and making a bad continuation bet. Plus if the BB has QQ,JJ,TT you may induce him to make a bad fold later on in the hand.

TURN: Given how you played the flop, and the fact that you only have one opponent standing between you and this pot, you must bet this turn. Checking and taking a free card with your made hand is not an option in this decent sized pot. Plus there is still a chance the BB will fold a hand that has you beat like QQ,JJ,TT. But he cant make this bad fold if you dont bet.

I think you played this hand excellently postflop and that is much more important than whether your A9s preflop call is right or not.