We rented a car and drove around the island. There is a car rental agency offering special rates right at the dock pavilion. Roads are easy and scenery is grand. We had to wait for a herd of cows to trot down one road. We didn't know cows could move that fast.
Weather was cold and wet the day we were in port. There was a shuttle bus that, for $7, ran in a circuit from the castle to the dock. Convenient but time-sucking, as there was a 30-minute wait between buses. The castle is interesting but difficult in bad weather.
Copenhagen is one of the best-run cities we have visited. From the cruise dock, a 15-minute walk takes you to an S train station, which runs straight into downtown/oldtown Copenhagen. The metro system is also efficient and runs to the airport.
In Copenhagen, when a pedestrian steps into an intersection, cars stop. Bikes don't, but cars do. Bike lanes are sacred territory.
The National Museum in Copenhagen is superb: award-winning organization, cafe with wonderful and inventive food, and free admission. Go.