The 3 hotels run a free shuttle to and from the airport.
If you are backpacking and headed for Hulopoe Beach Park, jump on the shuttle to Manele Bay Hotel and just walk down to the beach.
Free shuttle between 3 hotels
The airport is small, so you will need to fly to either Honolulu or Maui before catching a smaller plane to Lanai. You could sail over on a sailboat, but that would be entirely on your own.
The hotels provide free shuttles to and from the airports, and to and from the Koele Lodge and Manele Bay Hotel. The only time this wouldn't work for you is if you're not staying at one of these resorts.
There are few rental cars available, and they cost a lot ($100 a day). There's also no place to drive around and see, so you might as well just go to the resort and stay there.