The closest driving range you'll find to Sydney's CBD, Moore Park Driving Range, park of the Moore Park Golf Club, is a multi-story land-ship of tiny white cannonballs. You can grab the bus down Anzac parade, or simply walk over from the local area (Surry Hills, Paddington, Redfern, Centennial Park) - they even have plenty of parking if you drive.
A great feature of Moore Park Driving Range is that it's undercover, so if it's raining you can still have a hit.
A bucket of 135 balls will cost you around $20-$25. Get in early if you want a good spot (or late, the driving range shuts at 11pm at night most nights).