You don't really need to arrange anything ahead of time. The beach is all public -- a day pass might help if you really need lounge chairs and umbrellas. It's easy to get to -- there's a local bus that runs with some regularity, but a cab is probably better to meet the ship's schedule. Besides Royal Palms, there's a "public" developed beach near the Governor's House. The Marriott is also a possibility. The beach at the Westin is lovely, but not much for snorkeling. Better snorkeling at Smith's Cove and Cemetery Beach (unless you're really strong swimmers and can go out farther, not sure that works near the Westin, but it does some other places).
You might want to google the usual weather pattern for the time of your trip for a better guess about rain. And if the water's rough, it can be unpleasant to go out to the Stingray Sandbar (which gets pretty crowded when cruise ships are in -- but still can be an awesome experience).
Remember if you get some duty-free rum (Tortuga's Orange rum is delicious) you will need to have room to get it home in your checked baggage (we messed up on that once).
The Cayman dollar is $1.25 U.S. and stuff is generally spendier there.
