Hmm...I think the cruise we went on to the Bahamas stopped in Nassau on a Saturday, but it was a holiday so none of the shops were open. We took a water taxi to Paradise Island and went to the Atlantis and walked around, and then went to a nearby beach for the rest of the time. It was really nice!
I took a land trip to Nassau last winter and yes for the majority, stores were closed on Sunday. You can find souvenier shops open, but jewelers and most restaurants in the downtown Nassau area were closed. Of course the Straw market was open, but I'd pass on going there. I suggest, take the ferry over to Paradise Island for the day, Atlantis has a beautiful Casino, lots of shops and you can walk thru the reef which encompasses the hotel. It's a definite must see!