There is regular bus service via either Metro or Caribe Tours from Puerto Plata to Santo Domingo. The bus itself is under $15.00 US and takes about three and a half to four hours.
You would have to add taxi fares from the Puerto Plata airport to the bus station and from the bus station in Santo Domingo to your final destination.
The only flights between the cities are private charters (unless something new has been added recently).
As an alternative, you could take a taxi from the Puerto Plata airport to Santo Domingo. That would cost upwards of $200.00 US.