We've been there twice, once we took a taxi, the other we had cordinated a shuttle to get us there, either was fine and reasonably priced. Great hotel. We were there during a hurrican several years ago and the staff turned it into a great time.
Not sure about the route to the Hotel, but there is a mall center right by the harbor where the Cruise ships port with a Hard Rock and some really good local places on the backside (not facing the ocean). I have not been in a few years, I know some places were damaged in hurricanes since Katrina. Swimming with the Sting Rays was my favorite at Cayman. Have fun!