I like playing heads up, my information here might be way off though.  But this is how I mentally work this situation and in blind vs blind the situations are similar.  This will let you know how I think, but it does not mean it is correct.  So read with caution.

We did not raise pre, and we called his raise after we limped.

This says that we have a hand that needs to see a flop to win.  Hands with an ace in them tend to raise here unless trapping, and when you just call his raise to your limp you are really under representing your hand.  To be honest I would have probably made a big raise pre with such a hand as this is going to stink if we hit an ace and the pot gets big.

When we check the flop he knows that we do not have an ace, so he bets.  Again I much prefer to lead out, I knew this was going to happen, why didn’t I raise pre? Now I am going to get in all sorts of problems.

When we call the flop he knows that we will do this with weak hands heads up, and so he fires again because if we have bottom pair we might decide to find the fold.  Damn it I should have raised pre, at least I hope I can play the river right if I call. This is how I play heads up, it is full of these situations.  I am constantly thinking what I am representing and does the other players actions fit to what I am representing.  I call here. This will represent middle pair a lot of the time.  

Now on to the river, which we needed to think about before we make that call on the turn.  If he fires again, at pot, I fold.  He is last to act, and that in itself gives him a lot of check behind for value on the river, and we could be slow playing something huge to his big bets.  If he bets very small, then I call. Everything between the two sizes of pot and very small just comes down to what do you think he has vs what do you think he thinks you have.  But in general, I lean a lot more towards folding the river. A lot more toward folding. Damn it, limp call check call check call check, I hate it, why oh why didn’t I raise pre and lead the flop?

Now in a sense this is bad, because calling a big turn bet is generally bad if you are not going to call the river.  But this is heads up effectively, and maybe he knows that and so he can make a big turn bet.  But do not really do this in any other situation, because it would be bad as people tend to bet the turn a bit tighter in normal play than in heads up and blind vs blind.  The reason for this is often no one has anything in heads up and so people bet the turn more, where as in normal play people tend to have something more often and so check the turn a bit more. 

Another line here is to call this turn and then make a really small bet on the river no matter what hits.  But I generally would save that for missed draws with 7 high or something like that, but here you have an ace, not only that it hit.  It is like a set. I am not folding the turn, and I hope he checks back the river or does a small bet I can justify calling, other than that well played him and move on to the next hand with a river fold.  Lesson learned an next time I raise my ace against him big and then act more aggressive on the flop.  Raise pre. Bigger the better.

And yes, you have to think about all of this stuff when you play heads up and there is a ton of room for error so there might well be some errors in what I wrote.  So, raise pre and everything gets easier…   Did I mention at all that you should raise pre?