Ordrupgaard is a beautiful museum with lots of interesting exhibitions. Right next to the museum Finn jul's house is located, a famous danish designer and architect. 

Contemporary Art Gallery in the center of Copenhagen focusing on mainly international, abstract, minimalistic art. 

Natural wine. Great costumer service

Copenhagen Meetpacking district is a good place to go out for a beer. Duing the summer the hole place is packed with young people. There is several bars, galleries and restaurants in the area.

A Kilometer stone (circa 1925) stands directly outside Nørreport station, in Copenhagen K. It's easy to pass by, but definitely worth noting if you're in the area — The typographic work on it is incredible.