There are two options, both roughly equidistant and neither an obvious winner:
Japan is about 10 hours from Sydney, with direct flights to Tokyo. While snow is rare in Tokyo itself, there is tons of the white stuff anywhere in the Japan Alps (eg. Nagano), on the Japan Sea coast (eg. Kanazawa) and up in Hokkaido, where ski bunny haven Niseko has become a virtual Aussie enclave. Japan is obviously not Christian, but Christmas is celebrated as a consumer holiday along the lines of Valentine's Day, with plenty of Christmas displays in shopping malls, Colonel Sanders decked out as Santa at KFC, etc.
Korea is only a smidgen further away, with direct Sydney-Seoul flights also lasting around 10 hours. Snow in Seoul is more likely than in Tokyo, albeit far from guaranteed, and there's lots of it in the mountains, eg. 2018 Winter Olympics host Pyeongchang. What's more, a solid 30% of Korea is Christian and Christmas Day is a public holiday.
Direct flights to both hover around A$1000 return, with cheaper options to both available if you're willing to route through a third country and/or use LCCs. Jetstar offers one-hop flights to Japan via Melbourne, Cairns and the Gold Coast, while Air Asia flies to various destinations in both Japan and Korea via Kuala Lumpur.