I am sure it can be done, everything is doable in the Kingdom.
I know you can catch a train from the apt to Ayutthaya, takes 30 minutes.
Also know you can get a transfer ranging from minibus to car with driver from the apt to Pattaya. If i were you, and you are not enlightened here, I would contact one of the apt transfer companies and ask for a quote on the transfer. Secondly I would get a quote from an Ayutthaya cabs who will take you half way and then they swap you over with a Pattaya cab coming the other way. Yes they do amazing things and I often wonder what a first time traveller must think when the cab stops in the middle of nowhere, on the side of the road opposite another.
Have a great holiday, yan19454, sorry i don't have more precise info for you.
I'm just reinforcing what jpatokal said--use the buses. From both places to Bangkok-Mor Chit, buses run about every half hour. That's a really long distance for a taxi unless you don't care about the cost. The buses are pretty comfortable, too.