The closest airport to Koh Phangan is Koh Samui (domestic flight from Bangkok or Phuket).
As soon as you land at Koh Samui airport:
You can use the Lomprayah ferry company (https://www.lomprayah.com). You will get a joint ticket (car transfer and boat) to Koh Phangan, Thong Sala pier.
Please be sure your flight arrival is at least 1 hour before the boat departure.
The sea trip takes around 30 minutes (depends on weather conditions).
Once in Koh Phangan (Thong Sala pier), take a taxi and In 10 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)