Eat a Fergburger in Queenstown.
Ride the gondola up Queenstown Hill and enjoy the scenery from above.
Hike or bicycle the limitless trails and enjoy the best scenery around in the mountains surrounding Queenstown.
Swim with Dolphins in Kaikoura.
Whitewater raft in the Rangitata River.
Skydive in Abel Tasman National Park.
Do one of New Zealand's listed Great Hikes, such as the Tongariro National Park.
Do tramping on one of these listed Great Hikes, and stay at the community huts. Share a group bunk bed with 20-30 other strangers.
Visit the Pancake Rocks and blowholes at Punakaiki.
Do a full day Ice Climb of Fox Glacier with a ice climbing outfit.
Camp at Lake Pukaki and enjoy one of the best lake views of the island.
Mentioned in this answer:
If you have the time, take a day-trip from Queenstown to Milford Track. There are several bus tours each day, but make sure you find a glass-top bus with lots of stops. The ride itself, before you even reach the sound, is stunning.