There is a sticky named something like Punta Cana airport taxi fares here on DR1 - just do a search. But, it goes back to 2004 and wanders all over the place - I don't know if you will find what you are looking for. But, I know from personal experience:
$100 USD taxi PC to La Romana. More or less. I think they have a set list of charges at the airport. But, it may also depend on how desperate you look at 9:30 at night. The farther away from the exit door at the airport you get, the cheaper it gets, but the drivers also get more sketchy.
People are probably going to give you advice to stay over night in PC and take the bus the next day. This will be marginally cheaper if you stay in a cheap hotel in PC maybe $60 if you an find one - it is high season. But, then there is the taxi cost of going from airport to hotel and finding bus next day, adds to the cost. Unless you take motorcycle taxi (moto concho) - cheaper but potentially deadly at night. If you are used to this kind of transportation, no problem I guess. Depends on distance to cheap hotel. I can't think of one close to airport.
Express bus PC to La Romana is about $6 USD. 1.5 hours on the bus. Bus won't be running at 9:30 at night.
All things considered, the taxi from the airport near exit door to La Romana is probably the best. (Unless you can find the resort transport as PIB says above.) Choices depend on your tolerance for cost and inconvenience. Taxi avoids all but cost. No fun going at 9:30 at night though. I don't like the autopista at night (big highway to La Romana), but that's just me.
If you are inexperienced, don't like uncertainty, or potentially risky situations, and have the money, stay in the Sheraton near PC airport overnight and take the bus the next day.