Using the Ski Bus is a breeze.
If you are a guest on holiday in the region:
Simply grab your Welcome Card (which you’ll receive when you check into your hotel) and your ski gear, and head to one of the designated bus stops around the city. Departure times are available at your hotel’s reception, so you'll always know when to head out to catch your ride. The bus is free, but be sure to make your group reservation early (via your hotel reception) if you’re traveling with a larger party.
If you are a local and/or don't have a Welcome Card:
You can use the ski bus lines 412 to Axamer Lizum, 420 to Kühtai, 414 to Muttereralm and J to Nordkette and Patscherkofel for free IF you are in winter sports clothes and/or with winter sports equipment and/or have a lift pass for the respective lift.
However please check the specific bus lines and requirements for free travel here: https://www.innsbruck.info/getPdf.php?file=pdf/skibus_plan.pdf
Here is the ski bus Route Map, and here the time table, for a closer look!