28% of hands calling a raise is reasonable for a 53/6 but I can see more offsuit aces in his range: I can't picture him laying down A9o or A8o, or maybe not even any Axo against a CO raise. I have no problem keeping the big pairs in there although maybe we should discount them a little.

If I add all the offsuit aces to his flop calling range and I remove 99, then your equity becomes only 40%. The best possible draw on the board is a gutshot, which you have yourself, so yes I think this one is a case where checking back the flop is better (but then I am an unworthy 2NLer who checks back his set of 4's on the river, so what do I know?). You have a good hand with second pair and a gutshot to go with it, and you have position, so it would be bad to be forced to fold to a serious raise. So check back and see what happens on the turn.

It would be useful to have some indication of what his postflop tendencies are. Have you seen him raise before? Hard to say what his range is after his raise but definitely all the aces, sets, two pairs are in there, and possibly all pair and gutshot combos. Let's exclude gutshots alone and second and third pairs with no kicker to be conservative and we get: TT+,A2s+,KTs+,QTs+,JTs,A2o+,KTo+,QTo+,JTo against which you still have 36% equity. Since the pot odds are about 22%, I think calling is OK. Calling is not unlikely to give you two free cards to improve, since he might be wary of a better kicker if he has an Ax for example. You are also likely to win more if you hit, and he is not very likely to get much more out of you if you don't improve.

I don't think the Ac and his check on the turn change his range. Maybe he was trying to improve and he did not, maybe his hand is not that good and he waits to see what you will do or maybe he has a super good hand like a set (now a full house) or trip aces that he is slowplaying. So checking the turn is right.

Then he shoves the river and I can't see him do that without at least a pair of aces, unless he is tilting or bluffing. Even if he does that with his entire flop and turn range, you only have 26.4% equity on this board, and he is laying 32.8% pot odds, so you can't call.

I do this as much as an exercise for myself as to hopefully give something back to people who comment on my hands when I post them, so feel free to criticize what I said, it helps me as well.