I live in Tromsø. The weather here is very varaiable and unpredictable. In mid june to early july you may have nice weather (sunny 20 deg.cel) to not so nice (0 deg.cel rain) and everything in between. It rarely snows during this time, but you never know.
Your 2 hour excursion will cover the major tourist sights in Tromsø (Ishavs-cathedral, mountain cable car and a couple of museums). If you have got only one day exploring Tromsø I would concentrate on seeing more of the city centre. You should stop at "Skarven":
http://www.skarven.no/en/ It is a very popular bar/restaurant/pub with lots of lokal color. (Try having dishes made of dried fish, or just the dried fish itself).
Arcitecturally the city is nothing short of a disaster. It was not damaged during the war but has had a rapid development from the 60's and to date. Together with a rather loose policy about the esthetics of new buildings leaves the city with a mixture of 200 year old wooden houses and ugly square office buildings. There are a qouple of exceptions: the already mentioned Ishavskatedralen and Skarven. The library wich is quite unique, and the Skansen restaurant some hundred meters north of the city centre. Ask the lokals where to go.
And finally two pieces of advice: 1: If you are told that the city centre is just a nice walk along the sea shore, just walk in that direction. Don't bother! Don't waste time on what is a 40min-1hr walk along a dusty ugly industrial area. (I see the cruise passengers walking this route every day of summer) Take a cab instead (10-15$).
2: If it is not cloudy; DO NOT GO TO BED! Put on a fair bit of clothes and spend the whole night on the upper deck of the boat. Enjoy the spectacular sun as it never sets.
A rather long post, but I really hope you enjoy your stay here and hope you are lucky with weather. It can be the most beautiful and exotic place you have ever visited.
Audun (and excuse my English)