Oslo is an expensive city, but it should be okay for backpacker as long as you couchsurf or go camping.
You cannot avoid transportation cost to/from the airport, and I will suggest you to arrive by Oslo’s Gardermoen airport because you will be able to use 24 hours ticket for all zones from this airport.
We spent $39 per day per person.
We didn’t go to any paid attraction, we camped for 3 nights in Ekeberg
And we shopped a lot as Oslo was our last stop. We bought curtains, linens and other stuff from IKEA. (And the flea market too). If you don’t plan to do shopping, you can do it in $29 per day.
So in addition to backpacking price, each of us spent $31 extra, and so our daily expenditure becomes $39 per day.
Oslo was actually okay for us as there were many supermarkets around. We were eating salmon and shrimps almost all the time. And berries too (finally it was in season!)
We were pretty happy buying things.
If you have any question about saving transportation cost in Oslo, feel free to ask me!
Transportation cost was something that I calculated for a long time. And it worked out exactly the way I calculated. Although it cost us $14 day, it was the cheapest it could be. (It is Norway!)
And well, if you go to museum instead of shopping, than you will probably spend around $40 per day like we did!