1) Continuation bets are required of you regardless of situation, unless you are honestly scared of the flop or are caught completely bluffing, or meet strong resistance.
In this situation you aren't really even c-betting because you made top pair. The only two reasons, (obviously, I know), that you would be called is if he doesn't believe you or also has an ace.
If he called your preflop raise and didn't raise you himself, you can put him on less than AJ if he made contact. A good player is able to lay down A9 and under, so put the pressure on him. If he calls the cbet on the flop and raises or check raises on the turn/river, you might have to slow down a second.
In the end it's all situation. He might think you're stealing too much and pretending he has a hand thinking you have nothing, or he actually has an ace. Either way he probably wouldn't bet into you unless his kicker was paired.
Remember to read every part of the situation. Since he's BB he would act first, so if he's checking the ace on the flop, he probably doesn't have one, especially if he's only calling your raise and not ck raising. (or he has a monster...but that's poker).