Look into chartering a plane from Santo Domingo to Las Terrenas. There is a small airstrip there. It doesn't have lights, so you would need to land while it is still light. You didn't say how many in your party, but if there are several, it shouldn't be too expensive on a per person basis. Your other option if you must get there the same day is to take a taxi. Not sure of the cost, but I estimate around US$100.
Arriving that late in the day, hitchhiking should not be considered. Driving at night in this country is hazardous due to the drunks, cows, etc., on the highway. Consequently residents don't drive at night unless they have a compelling reason. You would be putting yourself at risk. I don't think you realize how far you must go.
Why not take a room for the night in Santo Domingo, see something of the city, and get one of the morning buses to Samana. Either Caribe Tour or Metro will take you there. Get off in Sanchez, the town before Samana, and take the transportation that meets the bus to transport people like yourself going to Las Terrenas.
If you decide to stay overnight, the Board can suggest places to stay that aren't expensive.