score:5
Ryanair flies to Oslo and Haugesund. From there it is still 1500, respectively 2000 kilometres to Tromsø.
Berlin to Oslo is 2000 kilometres as the crow flies and 2700 over land. The fast option is flying. Just have a look at the usual search engines to learn about the options.
The train is probably not much cheaper. Only much slower. You will have to travel through Hamburg, Copenhagen, Stockholm and Narvik. The last leg will be by bus. Train schedules can be found via the website of the German Railways (DB).
However, as you have a railpass it becomes cheaper. You only have to pay the reservation fees for the X2000 in Sweden, the extras for the sleepers and the bus for the final leg. The whole trip will last about 40 hours.
You may find some useful additional information on Seat61.
Upvote:2
If however, you would like to see some scenery on the way, going by a mode of transportation where you can dictate stops yourself is essential. Biking seems pretty popular in Norway, but because that seems tough, I'd just pick car. Doing so means that it will become a multi-day journey, but can be rewarding as well. Note that there is not a whole lot of scenery in flatland Sweden, so if you go for the looks, I recommend choosing non-trunk roads through southwest Norway. (Example to enter in GoogleMaps's routing: Berlin–Kristiansand–Rjukan–Tromsø.)
Upvote:5
You should be aware of the distances involved. By far the easiest way to get there is to fly. Use the travel planner of your choice, and check the websites of Norwegian (a somewhat low-cost airline with less draconian luggage practices than Ryanair) and SAS (the flag carrier) to find out when it is cheapest to travel (apparently there are no direct flights from Berlin to Tromsø). If you have a rail pass, you could also consider taking the train to Stockholm or Oslo and fly from there. The Tromsø airport website has a list of direct flight connections.
By train there are essentially two options: to Oslo (by ferry from Copenhagen, Kiel, or Fredrikshavn, or by train, changing in (at least) Gothenburg) and then onwards to Trondheim (ca 6h), change trains, to Fauske near Bodø (ca 10h), change to bus, and then a ca 12 hour bus trip (seems long, Google maps says 7h driving time, might be less than 12h. But includes a ferry connection). Or to Stockholm, onward to Narvik (ca 13h) (nearly all the train travel is inside Sweden, Narvik is not directly connected to the rest of the Norwegian rail network), and then around 4h by bus to Tromsø. The connections might not match up nicely.
Some brief information from the Tromsø tourist information: http://www.visittromso.no/en/Articles/Tromso_useful-info/Travel-by-bus-and-train/