Look here:
http://www.visitoslo.com/en/activities-and-attractions/oslo-pass/
Museums:
http://www.visitoslo.com/en/activities-and-attractions/oslo-pass/museums/
In the city center at the fjord vis a vis the railway station you have:
The Opera building. Nice to visit for a good view over the fjord. Possible
to climb up the ramp to the roof for the best view.
http://en.wikipedia.org/wiki/Oslo_Opera_House#/media/File:Full_Opera_by_night.jpg
In the city center (between the Royal castle and the railway station) you have the:
The National Gallery
Historical museum
At Frogner (10 minutes with subway, tram or bus) you find:
The Vigeland Park. Nice park to walk in and lots of statues made by Gustav Vigeland.
At Bygdøy (15 minutes with bus or ferry) you have:
Fram museum (Roald Amundsen's ship that sailed to Antarctica)
Kontiki museum (Thor Heyerdahl's raft crossing the Pacific)
Viking ship museum
Norwegian Folk museum (many old buildings from different centuries). Quite nice actually.
Norwegian Maritime museum
Akershus castle is nice to visit. Good view over the fjord and a nice park. Within walking distance of the city centre.
The Armed Forces museum is there if you like to see how WW2 affected Norway.
At Tøyen (10 minutes with subway) you have museums like:
Munch museum (art by Edvard Munch)
National history museum. Consists of the Zoological Museum, the Mineralogical-Geological Museum, the Botanical Garden and greenhouses
At Kjelsås (20 mimutes with bus or tram) you find:
Norwegian Museum of Science & Technology
At Holmenkollen (20 minutes with subway) you have:
Holmenkollen ski park
If you like to feel the city life of Oslo you can walk from the railway station towards the royal castle along the main street, Karl Johans Gate. Here you pass the parliament building and the national theatre. It's also close to the national gallery. Going left from the national theatre you pass the city hall and end up in a nice area called Aker Brygge.
At Aker brygge you can walk along the fjord and see the city life. You find lots of bars, restaurants and shops, but the prices are high. If the weather is good it's very nice to walk here. You have a good view towards the Akershus castle so it's possible to walk there as well. Akershus castle gives you the best view of the fjord.
Holmenkollen gives a great view of all of Oslo.
It's also possible to join ferry trips on the fjord with the Oslo Pass. Very nice if the weather is good and warm.
Weather forecast for 21-22 May is partly cloudy and 12-14 degrees Celsius. Not warm so a ferry trip on the fjord can be a bit cold.