Copenhagen is definitely worth a visit, even just a short one! Despite the fact that you just have a few hours before boarding your plane or before the business meeting, you can easily experience the city.
First of all, it is fast and easy to reach Copenhagen Central Station (Copenhagen H) from the airport - it only takes 12 minutes by train - and second the city is very compressed, therefore attractions are within short distance.
First stop on the tour could be the tourist information Copenhagen Right Now, situated across from the central station, where you can pick up a city map before heading out into the streets of Copenhagen.
Copenhagen by bus
A guided bus-tour is a good way to get an idea of the city. The tours last from 1 to 2½ hours and drive by or stop at some of the greater attractions of Copenhagen, like for example: The Little Mermaid, The National Museum, The Royal Palace: Amalienborg, the Parliament etc. The various tours all begin and end by Palace Hotel across from the Town Hall.
Through the city by boat
Another possibility to see as much of the city within a short timeframe, is to get on one of the canal tours. Quite a few of the sights and attractions in Copenhagen can be experienced from the seaside, and while the canal boats sail through the canals you get a chance to rest your tired feet and at the same time enjoy the attractions of Copenhagen.
The tours take approximately one hour and the Little Mermaid, The Parliament, The Opera House, The Royal Palace: Amalienborg etc. are some of the attractions the guided boats pass on the tour. The boats leave from the very picturesque Nyhavn. The canal area is very popular and packed with restaurants and cafés.
Copenhagen by foot
If you would rather experience the authentic atmosphere of the city a tour by foot is a great way to breathe in the atmosphere. From the town hall you should either go right or left from Strøget (the main pedestrian street); this is where the more interesting streets are situated.
To the right off Strøget is a street called Strædet, which consists of Kompagnistræde, Læderstræde etc. and to the left streets like Larsbjørnsstræde, Studiestræde, Sankt Peders Stræde etc. are part of an area referred to as the Latin Quarter. These streets have many quaint and ancient buildings that are several 100 years old and you will find many interesting shops; everything from Danish Design shops to small second hand clothing stores.
The Royal Theatre and The Round Tower
The walking tour through Strædet ends up at Kongens Nytorv, where the Royal Theatre lies. This is also where the picturesque Nyhavn is located with its many old houses and many restaurants and cafés that are situated with a view over the canal.
The walking tour to the left of Strøget ends up at Købmagergade, (another pedestrian street), close by The Round Tower. The tower was built as an observatory in 1642 during the reign of Christian IV and a 209-metre-long winding passage leads to the platform and the Observatory at the top of the tower - from here you have a magnificent view over the old town, house roofs and church towers.
If you end up at the area around Kongens Nytorv and need to get back to the airport after your Copenhagen-sightseeing-tour, the Metro will take you to Nørreport Station and from here the train back to the airport only takes 17 minutes.