Once when we were travelling through Sydney, we stayed at a huge YHA hostel near Chinatown. Cheap, cheerful and convenient.
YHA Central Sydney
Wotif is an excellent place, as suggested for hotel deals. As for the drive from Alice, yeah, it is a long one, but it depends on how much of the country you want to see. Flying in to Uluru would be wise for the time strapped traveller, and be sure to see the Olgas. They are all quite incredible.
If you do make it to Melbourne, make sure you do the drive down the Great Ocean Road.
Great Ocean Road