The city is good, but I lived in Nightcliff on Casuarina Drive for over a year and reckon there aren't many places better in Darwin. Its close to the airport, water, shops, pool, some of the best year round markets and its still pretty cheap to get a taxi back there after a hard night at the cav. (Oh, and crawling distance from the Arafura)
Most share places I saw there were about $100 - $130 a week, just avoid the bedsitters next to the pool.