Well, you won't have much time to do any additional activities, and be advised that restaurant food on Grand Cayman can get very pricey, even in casual restaurants. Local specialties tend to center around conch or turtle -- conch fritters, turtle soup, etc. I can't honestly recommend an eating spot. Not because there aren't any good ones, but because I haven't eaten there in more than a decade. (However, I was there last year. Just didn't eat there.)
Here are three quick options:
1. Roam around downtown Georgetown and stop in at whatever spot looks interesting. The funkier, the better, in my opinion.
2. If you're hankering for something a little more refined, hop a cab to the Grand Cayman Mariott resort for lunch on or near Seven-Mile Beach. The resort is 1.5 miles north of Georgetown. You could also try the Grand Cayman Hyatt. Both properties are nice, but bring $$.
3. Take a quick spin through downtown Georgetown (fairly typical Caribbean shopping fare) and return to the ship for what should be an unhurried and uncrowded meal that, let's face it, you've already paid for. ;-)
Enjoy, and safe travels.
cfurry