Sounds more like you have been dreaming instead of planning  which is good!
But we cannot decide for you. Either already get it in Australia, and perhaps need to get a new one in Ulan Ude if not on time, or don't bother at all and just get one in Ulan Ude when you are there.
Third option is to check for a visa application in South-Korea before you leave.
Next to all that, the winters are pretty though in Siberia and Mongolia. So at some point, you will need to move forward.
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2011 Overland from Amsterdam to Nepal and back
2018 Overland from Amsterdam to Tokyo via Central Asia and Mongolia
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