Standard rental agreements are usually a minimum three month lease before it goes on to a month by month basis, either party - landlord or tenant having a right to give a months notice to the other after the end of the three months.
Sydney will not be cheap on living including rental for something decent closer in to CBD or beachside.
If you want something that is not so bad a deal for travellers and can allow you to check Sydney out at a reasonable cost initially, have a look at
Sleeping With The Enemy or
Backpacker accommodation Sydney, Sydney Terraces, budget, cheap, hostel .
I still reckon you're making the travelling harder for yourselves as three months on from May, Sydney through July August and September and even into October can be pretty miserable weatherwise and it is the ideal time to be doing the top end - I'll be up that way in July myself actually, and there's far more to Australia than Cairns to Sydney.
Probably a better chance of finding work as you move about too, maybe even the pearling industry at Broome is not out of the question.
Anyway, if Sydney attracts and you get stuck there like a few thousand more, also check out
Sydney Classifieds - jobs in Sydney, flats to rent, Sydney dating as they have share ads and there's actually a web site called
Flatmates and housemates share accommodation in Australia as well.