May is getting into the start of tourist season...one of the best ways to sell a car like that is to tourists, so write up an ad and stick it up at backpackers and internet cafe's. There are a few on mitchell street (the main street in the city) that will let you post ads.
There is no specific 'car market' that I know of, but there is a few places where people often leave cars on the side of the road on weekends - Vanderlin Drive near Karama, and the Stuart Hightway on the way out to Palmerston are the two best ones that I know of, there is always heaps of cars there.
I know at one stage there was talk of owners being fined for parking their cars like that, but I don't know that it ever happens - there's usually heaps of cars in those spots. Just ask around, and a frienly local will definitely help you out!
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