it sounds like a romantic idea
for the budget you have youd need something like a VH registered Falco or Vans or some other high speed RAA type aircraft, and you want to be very confident with the engine.
I love flying light aircraft in Australia...its a great plce to fly.
You should have a look at the Casa website when you get here, and get ahold of their dvd on Australian weather conditions. There are a few classic weather patterns you need to be aware of.
Also, many of the arrangements for safety in outback flying relate to what you are going to do if you are forced down in the middle of nowhere, because theres an awfull lot of it out here........
406mhz beacons are compulsory, and learning how to use HF is important. We usually carry a satellete phone and lots of water also.
Be meticulous about avoiding dehydration in the cockpit in summer.
The kimberley in WA is agreat destination, as well as the Barrier Reef. You can rent very well maintained aircraft in Alice Springs...just impress Cliff with you preparation and airmanship and look after the aircraft.