Welcome to the Forum, and I hope your trip goes well. We went to Norway last fall, and college was a long time ago for us, but saving some $$$ is still important. We didn’t go all the way north to Trømso, but I understand that wi yet is when it’s most popular. I don’t know if that means it’s more expensive in January than at other times, but do some lodging price searches Online.
Are you considering dogsledding or other winter activities in Trømso? Again, look into those, to see if they will meet your budget.
We did partially use Norwegian Air to get around, to Trondheim, then from Trondheim to Bergen, and from Bergen to Oslo. They’re a “budget” airline, and especially if you’re traveling fast and light, with minimal luggage, can be surprisingly affordable. Quick flights can cover long distances (like to/from Trømso!) better, faster, and probably cheaper than some other options.
We had a rental car to drive around fjord country, renting in Bergen and returning it there, before flying to Oslo. It gave us flexibility, but check on costs. Some of the places we went aren’t accessible in winter. Also, I believe it’s even more expensive for younger drivers, like under 25, and so that might not be a practical possibility for you. If you do wind up renting a car, though, get an electric one - Cheaper than a gas-fueled car, and EV’s in Norway also get discounted prices for tolls and car ferries.
Search for hostels in the places you’re interested in visiting. Shop for food at grocery stores. In Norway, 7-11’s are regular supermarkets, not just convenience stores.
