I travel almost every 2 weeks from Santo Domingo to Punta Cana. There are 2 buses that give daily direct service with 2 stops in Santo Domingo and they drop you off wherever you want in Punta Cana/Bavaro. The cost is RD$210/person and they leave at 7am or 4pm, arriving about 3 1/2 hours later. The main bus stop (Expreso Bavaro) is located just off Ave. Maximo Gomez in front of the Nacional Supermarket (might not say much to you but a taxi driver would know). The second stop if the bus isn't full is in Parque Enriquillo but I wouldn't recommend it since it's a very crowded place, not tourist friendly and you get whatever seat is left over.
The other option is catching a bus from Parque Enriquillo to Higuey, then transfer from Higuey to Punta Cana. Buses leave continuously and if I'm not mistaken the cost is just a bit higher. I've never done it this way so I couldn't say how complicated it is but the employees are used to seeing tourists so they can usually help you out and make sure you get there ok.
As for the car, I really wouldn't recommend it unless you've driven in this country before. Plus it's a bit complicated to get to Punta Cana and so far I haven't seen any map that can get you there easily. Plus, if you're arriving at 3, that means you won't be on the road until at least 4:30 and in Punta Cana by 8:30-9:00pm. That means it'll be dark and you won't be able to see anything. Think twice about it. As for driving once at the resort, depends on how active you are. There isn't all that much to do around Punta Cana and everything is pretty far from one another...
My last recommendation would be that if you want to arrive in Punta Cana on the same day, pay the extra cash and get a taxi. As Ricktoronto mentioned, there are cream colored tourist taxis at the airport who speak some english and you should be able to get a reasonable rate and a comfortable ride. You'll get there faster, comfortably and if you need to stop, you can do it. If I had the cash, I'd do it this way.
Hope this helps. Enjoy your stay!