Tromso is a compact city that can easily be explored on foot. If you wish to take public transportation, the local bus system is reliable and covers most areas of the city. A single-ride ticket costs 37 NOK (~$4 USD), and can be purchased on board or through the mobile app. Taxis are also available, but can be expensive.
If you have more time, consider spending a night at the Aurora Sky Station for the ultimate Northern Lights experience. If you have less time, consider skipping the Aurora Sky Station and spending more time exploring Tromso's city center and local cuisine.