Two weeks at least. Ideally two years so you can experience Innsbruck through all the seasons.
It is like at any place in the world: there is so much to see and to experience that you won’t get to know everything during a short visit.
Most Innsbruck visitors stay for one day which definitely is way too short. To visit just the most important sights — still not all of them — you will need three days. If you plan your greet right at the beginning of your stay you get the benefit of many recommendations and insights tailored to your interests so you can make most of your visit.