The quickest way is by air, yet you are captive to the airline schedule and seating availability if you want to do a day trip.
Surface transport by rental car would be from Punta Cana - Santo Domingo to Nagua by the new toll road and Nagua to Puerto Plata and the return would be the reverse. figure 14 hrs round trip. It is feasible if driving is shared. Best bet would be to do one overnight in Puerto Plata. Motor coach would involve Punta Cana to Santo Domingo by Bavaro Express, then taxi to the Metro terminal in Santo Domingo, take Metro coach to Santiago and onward to Puerto Plata. 8 - 10 hrs one way. and then an overnight stay.
Regards,
PJT