If you don't fancy walking, Bike Mike runs three-hour cycling tours of the city, departing Sankt Peders Stræde 47 in the city centre, just east of Ørstedsparken (which is southwest of Nørreport Station). The tour cost includes bike, helmet rental and Mike himself, a great character with deep, attention-grabbing knowledge of the city. Cash only.
Note that tours must be booked online and in advance; the office is not usually manned, aside from bike pick-up and departure times. Seasonal and private tours are also available; see the website.