The closest airport to Koh Phangan is Koh Samui.
As soon as you land at Koh Samui airport:
You can use the Lomprayah ferry company (https://www.lomprayah.com). Υou will get a joint ticket (car transfer and boat) to Koh Phangan, Thong Sala pier.
The last boat leaves at 16:30. 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, a private car will be expecting to pick you up at the pier 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)