There are lots of shops in the Clock Tower shopping area within walking distance of where you will dock. There are free demonstrations of glass blowing from 9-noon and 1-4:30 at the Dockyard Glassworks. There is also a Bermuda Rum Cake Company at the dockyard. We just got back from Bermuda. The beaches are truly beautiful.
I agree with jodiradie that a bus ride would be nice, but not during the busy time when work and school are starting or stopping. I stood the whole way from Hamilton to the beach and back. The one bus wasn't so bad because it had high windows and I could at least look out. The other bus had shorter windows and I couldn't see the scenery. If you are going one place and then back again you can buy tokens for the round trip. But if you plan to get off and on more than once you should buy a day pass. Each ride cost $4 and a day pass is $12. The day pass is also good for the ferry across the bay to Hamilton and back. Personally I wouldn't bother with Hamilton since you have so short a time. You can get the same things at the Royal Naval Dockyards (except seeing the cathedral in Hamilton).
Another option would be to take the ferry to St. George's, the town on the other side of Bermuda. That would give you a mini tour of Bermuda. St. George's is a very nice old town with a very British feel. You may pass the same places on the ship on the way to the R.N.D.