Nyhavn waterfront as seen in October 2016
Regardless of the weather I would advise to head to historical waterfront district Nyhavn. The area itself is beautiful surrounded by colorful houses and historical wooden ships. The oldest house you can find in there dates as far as from 1681. Back in the past Nyhavn was a place where sailors came back and it was split into more decent part and other not so decent where prostitution was widespread. If rain gets really heavy you can escape in one of the bars or restaurants in the area and you won’t pay as much for having a drink or dinner than you would normally do in the touristy area. A typical two course meal will set you back around 250-300 DKK.
National Museum of Denmark (Nationalmuseet in Danish)
You can escape the rain in one the museums and one of them which is pretty good is National Museum of Denmark – Nationalmuseet. In the museum you will go back through time the history of Denmark. The museum is arranged in the way that the lower floors go back to the beginning of history and ends with modern times on the top floors. You will see things like working medieval clock, mummies, period of vikings and mock ship, skeletons and bones of the animals and pre-historic people, different room interiors in the way Danish people lived and dressed in the Middle Ages and Renaissance and so on. Allow about 2 hours for a visit, there is lots to see. I had just about an hour before catching a flight back to Dublin and just about managing to see most of it. Please be aware that most museum in Copenhagen are closed on Mondays. The entry day ticket currently costs DKK75. The museum also features a restaurant and a shop.
University of Copenhagen Botanical Garden
27 glasshouses will certainly shelter you from some rain and keep you occupied for some time. The complex is arranged in different sections such as Danish plants, perennial plants, rock gardens and many more. The most notable is the old Palm House from 1874 that is 16 meters tall and has narrow, cast-iron spiral stairs leading to a passageway at the top. Please be aware that opening times between greenhouses may vary. The entrance is free except when some of the events take place.