We'll be in Innsbruck for three nights in December.

I'm trying to figure out how much we can realistically fit into a short winter day and whether or not buying the Innsbruck Card is a good option for us.

Is it possible (and advisable) to take the Nordkeetenbahnen, which I understand goes to Hafelekar, AND the Hungerburgbahn to visit the zoo in the same day?

Also, I'm interested in the Christmas in the mountains event in Igls the following evening, but I'm not clear on how to get there. I understand there is a bus and a tram? Is one option better than the other?

The Hungerburgbahn is a railway that runs from the city to Hunberburg, which is the base of the Nordkette cable car that goes UP the mountain side. On a clear day it is well worth the trip UP the cable car even if you are not skiing.
The zoo is nothing remarkable - I would not go out of my way to visit that and miss out on something else.

Igls is not far at all from Innsbruck. Bus takes about 20min. Not sure how long the tram takes. You'll need to check schedules and stops - depending on your starting point.

Thank you very much J62. I thought the zoo would interest my niece who works at one and plans to study zoology. I see there's one in Salzburg too, where we'll also spend a few days. Would you recommend one zoo over the other?

I don't have any info about the Salzburg zoo, but if this is for your niece who works at a zoo then that is good enough reason to visit any zoo. If you contact them ahead of time who knows, but they may even give her a behind the scenes look, or put her to volunteer work for half a day...