grab the earliest Metro or Caribe Tours Bus down to Santo Domingo. in Santo Domingo at their station take a cabby to Parque Enriquillo, tell the cabby you want there the station for the Bus to Higuey. they depart there every 20 minutes all day long, modern comfy busses with A/C and WIFI running. in Higuey they drop you off at the Bavaro Bus station, where 3 lines depart. named Bavaro, Cabeza de Toro and Punta Cana. you take the bus for Punta Cana, departs every 20 minutes and drops you off right at the airport entrance.
the Bavaro Express Bus departs only around 4 times per day, if you miss the departure or luck out as they often sell the tickets for a ride ahead of time, you would be stuck in santo domingo with your flight departure approaching.
the Route via Higuey is the most effective, easy to use, comfy Busses and quick.
Mike