During the night, no buses operate away from the city. The cable car does not operate at night (except during summer). Bus 100 leaves for Setermoen at 21:00 (arriving 23:25), and you could get off at a dark spot, but a bus back won't depart before the morning, and you'll be very cold.
The last public transportation arriving to Tromsø is the southbound Hurtigruten from Skjervøy. On Tuesday or Thursday, you can take speedboat 3 to Skjervøy, 16:00 – 18:15, or on Sunday 14:45 – 16:55. Skjervøy is a small village and you could walk along the road until it's sufficiently dark, although 17:00-ish is a bit early for aurora-watching. Take Hurtigruten back to Tromsø at 19:45, arriving 23:45. On the Hurtigruten are good viewing conditions for the northern lights, as its mostly catered for tourists (from speedboat 3 viewing conditions are poor, it's a fast boat and going on-deck is not permitted, in fact you may have to fasten your seatbelts!). Note that Hurtigruten has the reputation to be expensive, but it's not too bad if you neither eat nor sleep on the boat (not more expensive than bus or regular boats).
Another, more outdoorsy alternative is to take a tent and spend the night somewhere outside. For example, I can recommend taking speedboat 3 to Arnøyhamn and walk for 20 minutes or so into the valley heading north from the fjord. If you do so on Monday or Wednesday, you can head back already the next morning, otherwise you're in for a longer stay (study the timetable). I was there last September, it's a very pretty place. You could combine it with some skiing if that's your thing, but bring some warm clothes.
In general, public transportation timetables (except for Hurtigruten!) can be found at http://www.tromskortet.no.