Like every place spoiled by nature and easy to access, Koh Nang Yuan is very busy. Between the people who rent the a longtail boat to get there and the tours or excursions that come from Koh Tao, Koh Phangan or Koh Samui, it is very hard to see the island for with only a small crowd. And obviously, in high season (from November to March), there are many many people! The best is to go there juste after the opening. Because it is a private island that opens to the public only from 10 am to 5 pm. Another solution to really enjoy it almost without anyone is sleeping in one of the bungalows of the resort installed on Koh Nangyuan island(book here). Just know that if it is pleasant in the morning before 10 am and in the evening after 5 pm when the tourists are gone, during the day, when they are there, you will probably want to flee the crowd and go to spend the day on Koh Tao.
It is forbidden to enter the islands with plastic bottles.