The Worst Journey in the World by Apsley Cherry-Garrard (1922) is not about Tromsø specifically, but it is the definitive account of Antarctic polar exploration — the tradition that sent expeditions through Tromsø and shaped the city's identity. Reading it before visiting the Polar Museum gives the exhibits a different weight.
Knut Hamsun's Growth of the Soil (1917) is set in northern Norway's hinterland — the same landscape of forest, silence, and winter darkness that surrounds Tromsø. Hamsun is a complicated figure (he collaborated with the Nazis), but this novel captures something real about the northern Norwegian relationship with land and isolation.
Miss Smilla's Feeling for Snow by Peter Høeg (1992) is a Danish novel, not Norwegian, but it is the Arctic thriller that everyone seems to read before or after travelling to the far north. The Copenhagen setting gives way to Greenland, but the sense of ice and darkness is applicable throughout the polar world.