You can tan on the island all year long, just go around I think 10am-4pm? And hang out there a few minutes (about 15ish) everyday and you'll start tanning.
You only need to spend 15 minutes per day on the island to go up a tan level, and maintain it.
I always do it.
I can't stand that sickly, pale white face.
Makes one look like a redneck.
Hopefully in the next one we're a bit more free to change that stuff at character creation