I don't know what your budget is in Copenhagen. In general it is an expensive city. We stayed 2 nights pre-cruise at The Best Western Hebron. It is a 5 min walk to the railway station and the Tivoli. We found it to be a good deal at 265 euros for the 2 nights including breakfast and 2 free tickets to the Tivoli. The hotel only has a breakfast room, no restaurant, but there are plenty of places to eat round about. It is on the edge of the 'red light' district, but apart from the 'up-market' gentleman's club in the same street we saw nothing of it. The room was very comfortable, the staff friendly and the breakfast superb. It was a hot and cold buffet and no restriction on how much you ate. There were quite a few business people staying there which we thought was an indication of the quality. There is a lovely courtyard at the centre of the hotel and complementary tea,coffee and chocolate. No bar, but a drinks machine in the lobby.
The other hotel we used post-cruise was the Hilton at the airport. Again a great deal as it was a Sunday night. Great place and you can walk to check in as it joins on to Terminal 3.