Low budget travel and Norway can be a tricky combination but it certainly is doable with a bit of planning.
However, I'm not sure I agree that Tromsø is the best place to see the lights, the wet weather at the coast means it is often cloudy. Abisko in Sweden, just across the border, is usually a better option for the lights. (This blogpost does a pretty good job of explaining why: https://www.heartmybackpack.com/blog/best-northern-lights-europe-abisko/ .) There are tour companies in Tromsø that will take you to a spot with no or less clouds, but they can drive pretty far. One of those tours can mean 2 hours in a minibus out to somewhere, and then 2 hours drive back.
If the goal is to see the lights and nothing more I recommend December, it is dark but not as much snow yet and not as cold as January or February. But if you want to do other winter activities, January is usually better as there is a bit more snow.
Ryanair has very limited presence in Norway, only a handful of flights to Oslo and Sandefjord (which they call Oslo in their marketing) if I'm not mistaken. But there are other airlines serving the area. And there are some World war II-related things to see in Norway.