We stayed at the Crowne Plaza in Copenhagen. As it turned out, this is the hotel used by NCL for passengers booking accommodation through NCL. It is brand new, very European, and quite comfortable. It is quite a ways out of the town but both the Metro and the train stop 100 metres or so from the back door, so that is not much of a problem. There is also a shopping complex next door (closed when we were there on a Sunday). The distance is only a factor when you are going to the ship. If you booked your accommodation through NCL, it is not a problem as a shuttle is included. For the rest of us, however, it was an expensive taxi ride. We found out after the fact that we could have taken the train to a stop within walking distance of the ship. That is what we used on our return to get to the airport.
We arrived a day early to see Copenhagen. What a pretty town. So many bicycles. Walking down the pedestrian mall (largest in the world) is worth the experience. We had dinner at a restaurant called "Det Lille Apothek" (The Little Chemist). Highly recommended. Make sure you wander down to Nyhavn and enjoy a brew, lunch, sandwich, glass of wine, whatever, outside soaking up the atmosphere.
The ports on the cruise were fabulous. So much to see and do. We were exhausted mentally and physically at times.
We used spB Tours for our shore excursions in Berlin, Tallinn, St Petersburg and Helsinki. Excellent guides in Berlin and Tallinn especially and very good guides in St Petersburg and Helsinki as well. We would not hesitate to recommend them (spB Tours) to anyone. Excellent value for money.
We recommend that you take the boat cruise and wander around the market near where the boats depart. The Old Town is well worth a half day wandering around. Try to catch the changing of the guard at the Palace.