Just to expand on what Chuck said about not calling 4bets preflop:
When you call here, you're essentially hoping that he's bluffing
or that you're in a coinflip situation. And while JJ might be in his range (far from everyone 4bets that preflop), that's only a small part of his value range and, worse, you don't know whether he has JJ or KK. And not only that, but when you flat the 4-bet with the intention of fit/folding, you'll only pair up one time in three. This puts you in a situation where you call $33 and need to win ~$100 on average the times you flop a pair.
This would be fine, if you know he has JJ. But he's going to have KK as often as JJ, and suddenly half of your pairs are going to cost you your stack, and there's no guarantee that KK (or JJ for that matter) will pay you off if you flop an ace. And I think you'll find that the majority of the times you get stacks in on an ace-high flop, he will have AK.
So why is shoving better if you think he's bluffing often? Because AQ is not that big of a favorite even versus most bluffs, and the fold equity you get from shoving will show a profit. And if he's not bluffing often? Then folding is automatic. You're at best putting in a third of your stack preflop as a 44% dog.
You can flat 4-bets sometimes with aces or kings in position, but you shouldn't with a hand like AQ or AK. I have a really hard time coming up with ways for you to make money postflop. The fact that they're suited is not much of a consolation because with the SPR you'll have on the flop your implied odds
(which suited hands thrive on) is going to be virtually nonexistant anyway.