there are tonnes of threads on getting to and from punta cana and santo domingo. just use the search bar at the top of the screen. i had the same question last month (but only for land transport) and got some really good feeback. check out that thread at:
Overnight Trips From Punta Cana to Santo Domingo??
if im not mistaken the express bus takes 3-4 hours. i havent tested this out yet as im not making my trip until next week (woohoo!!)
happy travelling!