Assuming the process for getting recognised by immigration and recognised by the airline is simple, just focus on making your online health record is up to date and linked to Medicare.
As soon you are in the country pop into a chemist who is a registered vaccine provider and ask them to add it to the AIR - the immunisation register. For me that bit took me <5 mins, I just went to the closest chemist (who are registered vaccine provider, but don't do COVID shots) at a quiet time, and they were happy to do it. The longest and most painful bit for me, having been away for 12+ years was getting my individual health identified (IHI) set up although if it's already set up would be much easier.