I visited Oslo on the cheap last October. A few things to note:
1. Food in Oslo can be expensive. However, Oslo's City Hall has a cafeteria in the basement (access through doors facing the fjord and train, not main entrance) where you can get a full, delicious meal for 40-60 kr. It is meant for government emplooyees but opens to the public Mon - Fri after 12:30 PM. The hot meals change each day, but there is a permanent salad bar. Great hidden jem.
2. Also, if you are looking for free bathrooms in that general vicinity, the Oslo City Hall has them (both near cafeteria and through the main entrance side) as do the Akershus Fortress (housed in its own building on the grounds).
3. Many national museums have a free day each Thursday. You can visit the National Museum of Art and see the famous painting "The Scream" among others!
4. Oslo City Hall and Akershus Fortress are worth seeing. The City Hall has beautifully decorated rooms and paintings and is free to the public (main entry is the north entrance opposite the side facing the train). Also, the grounds of Akershus Fortress are free to wander (museums and tours are extra).
5. Walk the roof of the Opera House. Yes, the roof. It is an architectural masterpiece and has several slanted walkways which offer majestic views of the city and Oslofjord.
Enjoy Oslo! 🙂