I'm sorry, but you've picked the worst possible time to find a bed in Sydney. It's gorgeous summertime, everybody's on holiday, and from all corners of the continent and the world, they all pile into Sydney to catch the New Year fireworks. Add in that there's demand outstrips supply all year around, meaning that even perfectly ordinary hotel rooms in the off season cost $300/night, and you get absolutely ridiculous prices -- looking at Wotif, eg. the famously cramped Holiday Inn Old Sydney off Circular Quay wants $1080 (!) on Dec 30th, and is sold out anyway on the 31st!
Your saving grace is Sydney's hostels, there are tons and they are cheap (from ~$20/night). But book now, since they're popular too, and be prepared for a hard-partying crowd if you're anywhere within striking distance of a beach, Bondi in particular is a notorious disaster area this time of year. I'd look around Central/Surry Hills, it's close to everything but a wee bit more laid back. George St/Chinatown, on the other side of the station, also has lots of options but it's a pretty packed area.