Your Own Country...

Do I have money? Or am I like Gilligan and Marianne?

Just ownership, and let's say tax-free (or up-to-date on taxes). With whatever you own now (bank assets, property, whatever you have now and can afford to buy now, etc.).

So, it just fell into your lap, tax-free (at least, at present).

Using my resources I would need to build a raft and hope I could navigate it there. Then I would live like Tom Hanks. Maybe not that. I likely would look to attract investors instead. Turn it into a gambling resort.

Sell it.

Will it be completely destroyed in the near future or relatively near future from effects of climate change?

That's a big deal breaker. Even corporations would probably not want to buy it for resort purposes if that may in the island's future.

If not? I'd try and build an animal sanctuary and a residence for scientists (including a small studio apartment for myself).

Build my own personal resort - then hire some peasants to serve me, massage my feet while peeling grades and feeding them to me.

Make it into a wildlife sanctuary.

Exactly my sentiments. I'd clear out any non-native species, lock the door, and throw away the key.

What's the hurricane risk where it is located?