When you say "Public Transportation" you can mean nice, big buses or not so big 30 pax buses that carry up to 40...Jarabacoa is serviced by the Quinto Patio line and you can get a direct bus to Santiago, you just have to ask.
Then either Caribe Tour or Metro will take you in safety and chilling comfort to Sos?a and then you can catch a local bus out to RSJ. It will be an interesting few hours and you you should have some fun. Keep smiling.
HB