The closest airport to Koh Phangan is Koh Samui (domestic flight from Bangkok or Phuket). Take a morning flight to Koh Samui so that you arrive on time to catch the noon boat.
You can pre-book your boat ticket (Pralarn pier Samui) at https://www.lomprayah.com. Ideally, you will arrive in Samui before 11:00 and take the 12:30 boat to Koh Phangan.
Alternatively, as soon as you land at Koh Samui airport:
Use the Lomprayah ferry company (https://www.lomprayah.com). You will get a joint ticket bus&ferry to Koh Phangan, Thong Sala pier (any other pier, the distance to the resort is longer).
Please be sure your flight arrival is at least 1 hour before the ferry departure.
The sea trip takes around 30 minutes (depends on weather conditions).
Once in Koh Phangan (Thong Sala pier), take a taxi and In 15 minutes you will arrive at the beach resort.
The whole trip from Samui airport to the resort may take around 2 hours (depending on the time of your arrival, ferry boat schedule and weather conditions). Total cost is around 600 baht ($20)