I will be going to Japan from Australia, and I am considering to buy a Dyson 360 to bring and use in Australia. Q: Would the Dyson 360 bought in Japan work with the power supplier in the box or would I need a transformer for it to work in Australia?
You should not expect this device to work "out of the box". The voltage of wall outlets in Japan is 100 volts, while the voltage of wall outlets in Australia is 230 volts. In effect, this means that a device built for Japanese homes will draw much more current than expected in Australia, which will cause it to put out much more heat than expected. Unless you're trying to start a fire, this is a Bad Idea™.
Note that the USA runs on a similar voltage (120 V) to Japan. So your plan will probably just replicate your experience with using a USA Dremel tool in Australia.
As an aside, the plugs are also of different shapes: Type A/B in Japan, Type I in Australia. This means you couldn't even plug the thing into the wall in Australia without having some kind of jerry-rigged connection, which is another Bad Idea™.