Teresa,
My family and I drove over to the Punta Cana/Bavaro area about a month and a half ago. My time from downtown Santo Domingo was around 3 hours, driving in the afternoon. If you are not aware, nighttime driving is quite stressful here and requires extra time especially if you are in an unknown area. Since the airport is on the Punta Cana/Bavaro side of the city, that whould knock off about 30 minutes from my time if conditions are the same. The road is undergoing expansion from a two lane to four lane for the first hour or so (don't remember exactly where to where), which did not create any problem for us. I know the bus lines Metro and Caribe tours have terminals in the Punta Cana/Bavaro area, but I'm not sure if they stop at the airport or if you need to go into the city. I hope this helps and have a nice trip.
Greg