You should probably book in advance, but more than 2–3 days is not needed unless you want an overnight trip.
There are two aspects regarding booking tours in advance:
- Tours might be fully booked
- Tours might only go if there are enough people
The anwer depends on the kind of tour. Tours vary from an extended taxi ride to a dark corner, to overnight tours spending the night in a cabin in the wild. If your aim is just to see the aurora borealis, you should be able to find a simple tour some 2-3 days in advance (close enough to consider the weather forecast). But if you want to spice it up with other experiences — a night in a cabin in the wild is an experience even if there won't be any aurora — then booking it longer in advance is recommended. You're describing that you're looking for a full day trip. But, aurora trips are at night and night doesn't start so early in March, so I'm not quite sure what you mean by that.
Another aspect is: how essential is the tour to your vacation? Last time I was in Tromsø, we went solely to do a winter whale safari, and we booked a long time in advance. But if you're just travelling around and considering aurora watching as one out of several activities, you can take a higher risk.
Disclosure: I was living in Kiruna (Sweden) 2008-2013, roughly 350-ish km from Tromsø, and Kiruna has comparable aurora tours for tourists. I also visited Tromsø on perhaps 6 occasions or so during this time, although I did not participate in any guided aurora tours (because I don't need a guide to see the aurora).