We go to Hokkaido every winter. My son started going when he was 2 years old too.
Yes, it will be cold, but most of the places have heat so it is not too much of a problem. My reminder would be should you be walking outdoor anywhere with snow on the side of the walk. DO NOT carry your child. Hold on to your child and walk or use your pram. Because, the road surface could be icy, and if you fall yourself then your child would fall too. It would hurt more falling from 5 feet height instead of 2 feet height.