Once upon a time, perhaps end of December 2005, I made the trip landing in PUJ late afternoon, almost evening, taking a cab to Higuey and staying overnight there. Up before dawn to get to the bus terminal for a 5:00 AM bus to The Capital. Then another taxi to Caribe Tours and a bus to Los Charamicos then a moto or taxi, I can't recall which, to Rocky's... for the New Year's Eve party. I arrived at 1:18 PM hours earlier than I was told it was even possible. It was not an easy journey and I would neither recommend it, nor do it again!
There was not enough time to reverse the trip, nor any inclination on my part, so I arranged to take a 10 seater turboprop from POP to PUJ very early in the morning the day of my return flight. It was $119 plus tax and was a short hop, 1 hour in the air. I then had a few hours to kill and PUJ was not a bad place to do it, at least back then.
My advice to all is fly into the airport closest to your destination. Any money you save will come at high cost for the inconvenience and wasted time, as well as the cost of transport. It's simply not worth it in most cases. In this case, if the OP insists on flying into PUJ then I'd highly recommend sucking it up and paying for a taxi. It'll be a few hundred USD more or less but worth it.