You'll likely find that most of the diveops do SRC in the early afternoon. Or too early in the morning for you to get off the ship in time. Unless it's changed very recently, all cruise passengers are tendered into Georgetown.
It's probably a 15-20min. drive to the marina in the North Sound where most of the boats are kept and then 30-40mins.? to SRC. It's a 45 min. dive so as long as you're in port till 5 or later it should be possible if you get on one of the 1PM trips.
Understand that Georgetown is typically a zoo when the cruise boats disembark (I"ve seen 3 ships moored at the same time) so the logistics of getting out of there might be more than it would appear on a map.
You might check with Deep Blue Divers, they're located by the Lobster Pot in Georgetown. IIRC you could walk there from the cruise port. They do SRC dives daily.
Scuba Diving Grand Cayman Dive Snorkel Cayman Island
I respectfully disagree with the above poster. SRC on scuba was a blast. You overweight and sit on the bottom while the Stingrays swim between you. Just about every nearby diveop does it at least weekly, some daily, they wouldn't be if it was disappointing. We had snorkelers on our boat also but the rays never interacted with them unless they stayed down for more than a few seconds. If you only snorkel, do the Sandbar trip instead, it's 4-5' deep so you can stand and interact with the rays.
We got mugged by several stingrays, I even had one sit on my head for a short while. The dive itself is in a sandy area with a couple of small coral clumps nearby so it's not much to look at once they leave. (when the squid is gone)