Anyone know about how much it would cost?

Why not take the train? There are tuk tuks for hire for the the day waiting for tourists.

I agree with dgunbug the train is a more attractive option although you will need to bargain to get a decent deal from the tuk tuk drivers. There are frequent trains between Bangkok and Ayuthaya. So you can be pretty flexible on timings. An even more attractive option might be be one of the tours boat/bike/car/ train or any combination thereof which will usually take in Bang Pa In

When you get into a lot of BKK taxis, there is a card hanging in the back seat. They advertise tourist trips. Ayutthaha should be listened. In looking at these cards in the past they are pretty standard taxi to taxi.

A boat trip is by far the most enjoyable

Bob