Your best options would be either Qantas or Japan Airlines, both of which fly directly from Tokyo-Narita to Sydney, from where you can connect onward to Melbourne or Brisbane. (There are no direct Tokyo-MEL/BNE flights on any carrier, even Jetstar requires a transfer at Gold Coast or Cairns.)
- Qantas: Children Travelling Alone
- JAL: Kids Travel Support Service
Per the policies above, both permit 6-year-olds to travel as unaccompanied minors and, If I'm reading those right, also allow flight connections (outside the US). However, given that int'l-to-dom connections at SYD are enough to make a grown man cry, that does sound kinda tough even for a well-traveled six-year-old.
All things considered, I'd plump for JAL, which in my experience -- although not with unaccompanied minors specifically -- bends over backwards to take care of kids.