If you just google "outdoor equipment rental Queenstown" you'll find plenty of places that will rent backpacks, sleeping bags, etc. Because of the excellent hut system, all you really need to bring is a backpack and sleeping bag, since the huts have mattresses, lighting, flush toilets, stoves, and clean water. I really wish they had a system like that here in US.

I highly recommend the Routeburn, easily one of, if not the best hike I've ever done.