score:5
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.)
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.
Update: Qantas started flying NRT-BNE direct a few times a week in 2015, and this is now clearly your best bet.
Upvote:5
It is possible to book flights on China Eastern, China Southern, Singapore Airlines, and Cathay Pacific, among others, to Melbourne, which all have unaccompanied minor services for children older than five years old (except Cathay, which starts at age six). As you might expect, there will be a fee for the service, which depends on the specific airline.
However, in none of these cases (as with Jetstar) are these flights nonstop services. Given the length of these flights (i.e. 13+ hours), you should carefully consider whether or not your six-year old will be fine flying without a familiar adult for that period of time, and the layover will do little to help things.