You need to get to Santo Domingo (about three hours), and then from Santo Domingo take the Caribe Tours to Sosua (four hours). From there, you can get a commuter bus to Rio San Juan (one hour depending on stops). If you go by public transport, should cost you around US$25.
I've been trying to get into Caribe Tours' website for over an hour now, to give you the schedules, but it won't open.
Here's the link and maybe it'll work for you later. http://caribetours.com.do/
If you don't understand Spanish and can't figure it out, let me know.
From Barahona to Santo Domingo, leaving at 6:15 9:45 AM & 1:45 5:15 PM.
Leaving Santo Domingo to Sosua at 6:00 7:00 8:00 9:00 10:00 11:00 AM &
12:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 PM.
My guess is that Barahona to Santo Domingo takes 3 1/2 hours, and 4 1/2 hours from Santo Domingo to Sosua, then figure on 1 1/2 hours of public transport to Rio San Juan.