Against a 37/4 you can't go wrong with betting big. Because of his wide range for calling your c bet the proportion of flushes will be low overall. I wouldn't expect him to be ever getting out of line with a worse hand hardly ever once he shoves. Maybe TT, but don't level yourself that he's doing this with a bare Ah with any kind of frequency.

As for calling it off, the problem here is that he's not full stacked. For 100bb it's a simple fold, but getting about 2.5-1 I think some will advocate calling. I still think I would make a fold, even with the Jh but it's close.

The way I tend to look at thse versus a player like this is, they are passive obviously and throwing chip at me, they're not bluffing and we beat A9o which may well have Ah giving villain some outs, but even that hand is unlikely since they do tend to play passively - and TT, that's about it really.
Sine I can't see our opponent doing this with TPTK very often it's a pretty nut orientated range and I'd make a tank/fold.

I'd be inclined to bet/fold the turn vs. a passive opponent and you have to think about stuff like that before you pick the sizing since going for 2$ puts you dangerously close to being priced in for the call.