You will not have a lot of time, and it is an expensive island. I may have missed a "must eat there" place, but I have not seen or heard of any, except at the "high end". If you are willing to pay, there is some great food on the island. I have not been to it yet, but the Ritz Carlton is getting rave reviews. The island has not fully recovered from the last hurricane, but they are making real progress.

Since you have just 1 day there and can do just one thing, I would take the Submarine ride or the trip to "Stingray City". Both can be bought on board. You can also buy them ashore after getting off the ship, but in the case of stingray tours, that is a bad idea. If you buy on board, you will go to Stingray City (a sandbar a few miles off shore) on a large, modern purpose built vessel. When you get there, large, sturdy steps drop down from the boat to the sandbar. You will be given food to feed to the rays, and you will get great memories. If you buy the tour ashore, you will end up on a rickity Capt Ron type boat, and you will provide entertainment to the folks on the nice boat as you struggle down a rope ladder thrown over the side. Both the Atlantis submarine ride and the Stingray City tours sell out early. Depending on the cruise line you can prebook the tours before you go, during the pre registration process.

The turtle farm tour (I have not been on it) has not been very interesting to my clients.
http://www.caymanislands.ky