http://freakyflier.com/2010/11/17/jumbo-hostel-arlanda-airport-stockholm-sweden/
very quick google search 
I see and raise your Jumbo with the af Chapman hostel in this pic:
Did work in the tourist business for 2-3 years in Stockholm, and mostly advised people to go see The Vasa/Wasa-ship (Vasaskeppet). The worlds best preserved 17th century ship (from 1628). Built as the most modern and war-ship under Gustavus Adolphus rule, meant to be the glory of the swedish fleet.
On its maiden-journey, it made it about 1 kilometer before it sank.
Now, as a kid living in Stockholm you take field trips there like every year from the age of 6, so seen it way too many times that it lost it's charm. However, for a first-timer (and history-buffs in here) it should be a great view. (back in the 80's you were allowed to go on board, not anymore).
Made a quick map
Vasamuseet is underlined with black. On the same island "Djurgården" there is a lot of museums, the liquor-museum, ABBA-museum, and what I would recommend: Skansen (with google-maps pin on it). An open air museum with a zoo as well. The zoo takes up most of the park, but many old 19 and 18th century classic swedish buildings - working as a museum. Just taking a stroll here can be great on nice day.
Underlined with pink, is The Modern Museum. A very good museum of modern art. with brown is the acclaimed photographic museum supposed to be really good - haven't been there myself.
Inside blue circle is the old town, with narrow streets, cozy restaurants in basements and a few nice bars.
The red lines are perfect walkways up on the cliffs, giving you a great view of the city - love them both when out walking.
Light green is Medborgarplatsen, a square with a few bars. Lots of room to sit outside, and a meeting place. Not remarkably noteworthy, but a great place to sit in the sun and drink beer.
Black dot (on Närkegatan) is my favourite bar in stockholm. In a basement there is a bar in a bohemic fashion, (well, hipster-style). With boule-courts, ping pong tables, board games, darts, some video-games. Combined with beer, great music, and nice people - this place is great IMO.
Dark green dot is a great place for lunch. Kafé 44. The staff aren't over-friendly, but that's part of its charm. Everything is cheap, it's a very left-winged-cafe so to speak. Everything is vegan. different lunches every day (but always several sandwiches and soups as well). The sell Cuba-cola, not coca cola or pepsi - due to the places left/anarchist-history and theme.
Hope I could be of any help, just a few tips.
Edit: Granted, I never really venture north of the old town. That's the central and posh part of town (especially Östermalm), a beer costs 7-8 €, people are judgemental and excluding. There are of course nice places there as well, but I kinda wouldn't know.
