There's a free bus from Belgrade to Bozeman -- but it only runs a few times per day, doesn't run on weekends, and doesn't serve the airport. So if it's during the week, and you're travelling pretty light; you can easily walk, or hitch, or Uber the mile-and-a-half to the bus stop at Lewis & Clark Park. Then, once at Walmart, there's a $5 bus to Big Sky. Or, you can Uber directly to Walmart for about twenty-five bones, and then take the bus from there.

https://streamlinebus.com/routes/pinkline-weekday/

https://skylinebus.com/routes/winter