If you want to do like the locals, go to a place like Ægir and get hammered. It's cheap, it's fun, you can usually count on decent music, except when you can't, and you'll meet someone friendly, guaranteed. It's definitely a must for experiencing Copenhagen.
A part of a dying breed, these "cafes" are slowly dying out. It's a bit of a time warp stepping in - you could swear nothing has changed in the past 20 years. And again, it's cheap. They sometimes keep the bar open past normal opening hours - depends on the crew.
Tu-Su: 16:00-02:00