Unlikely that you would find anything affordable at all in Sweden or Norway. For better advice, I would suggest you let us know what your time limit and budget would be? If you're not ticking off the box of riding all the way from home to Nordkapp, then a lot of the journey through Finland or Sweden would just be boring motorway anyway - and honestly there is a good chunk of central Norway that also goes into that category, unless you have basically unlimited time to follow the coastal route.

I've done Nordkapp twice on my bike, other parts of the Arctic even more times. Starting in Tallinn, I tell people to take the overnight car-train from Helsinki to Kolari - you have an afternoon's easy highway to ease you into the Arctic road conditions, the really interesting stuff starts around Enontekiö and the Norwegian border. You can ride from Kolari to Honnigsvag, the nearest town, in a single day (Midnight Sun, remember!), go to Nordkapp the next morning before the ticket-checkers start work, then start heading south via Alta towards Tromsö - from there using the ferries through Senja to Andenes at the tip of the Lofoten archipelago, down towards A/Moskenes and the ferry down to Bodö... south of Bodö/Saltstraumen it starts getting less impressive, definitely south of Mo i Rana you need to follow the slow coastal road for meaningful scenery, the inland road is just a boring day's highway to Trondheim. The interesting stuff starts again from the Atlantic Road, you can start there in the morning and make it over Trollstigen and down towards the nice area around Hardanger before getting to Oslo.

Minimum to get from Tallinn/Helsinki/Stockholm to Nordkapp and back, to check it off your list? Can be done in a four-day weekend with the car train. Or you can easily spend two weeks going through the length of Norway, slowly.

You can rent a travel bike out of Tallinn for >100 EUR per day.

EDIT: Given the cost of hiring a bike, I would actually look into shipping your bike from Budapest up to somewhere where you might want to start and end - could be Tromsö on the Norwegian side, could be Oulu or Rovaniemi as the last big towns on the Finnish side, or maybe even Kiruna in Sweden - anywhere that DSV or another big shipping company might have a terminal. Pack all your moto and camping gear along with the bike, then take a Ryanair flight up to meet it at the start of your holidays, pack it back up and fly home at the end. You will be without your bike at home for a week on either end, but you will save money, boredom, and wear&tear on the bike.