Okay, but know that you will need to know Spanish and not have a lot of baggage.
When you arrive at the Airport, go to curbside of the second floor departures area. Wait for a motorconcho (passenger carrying motorcylce) to come by.
Flag down the motorconcho. Take a ride on the motorconcho to the parada de los minibuses para el este en la cruce de las americas - the bus stop for buses going east at the intersection with the main highway. This will cost you about 40 pesos. Then wait for a bus with a "La Romana" placard in the winshield. Flag down this bus and take it to La Romana - maybe 60 pesos.
When you arrive in La Romana, you can ask where are the minibuses or publicos (fixed route taxis) to Bayahibe.
You should allow quite a few hours to do this all.