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Realistically it's going to turn into a tourist trap.
So you'll want an airport, a dock for cruise ships, a nice walking path right behind the beach, and maybe a nice ferry from the airport to the main tourist strip.
I'm assuming that the island is small enough for somebody to comfortably bicycle from one end to the other, so the only infrastructure that'll really be necessary is footpaths, bike lanes, and some roads so that deliveries can get made.
Make it a pain in the ass to get cars on the island, both so that traffic doesn't become an issue and so that tourists have to rely on buses or expensive cabs.
Thanks for the hypothetical OP, I love these.