What exactly were you quoted for - connecting flights to Hammerfest and then a bus? Connecting flights will often be expensive, it will probably be cheaper to break it down and buy the flights separately, as there can be competition on the segments.
Normally people on a budget getting to Nordkapp fly to Alta, and then take a bus from there. Hammerfest would be a closer airport, but it has only local flights, not even to Oslo, so is likely to be rather more expensive to get to. It may also be practical to take the bus from Tromso, though probably requires an overnight.
From Stockholm, the only useful flights to Norway for you are to Oslo or Tromso; the Tromso flight is summer only.
So consider the following, maybe they are cheaper:
(1) buy a flight to Stockholm to Oslo and a flight from Oslo to Alta, then bus
(2) buy a flight to Stockholm to Tromso (if dates make this possible, but there is no competition on this route so it may be expensive) then bus
(3) buy a flight to Stockholm Tromso then a flight from Tromso to Alta or Hammerfest, then bus
But as people say, Nordkapp is rather disappointing in itself, it isn't even the northernmost point of the piece of land it lies on. Visiting it makes more sense as part of a roadtrip to the region, so you can say you visited the northernmost point of the European road system.