Does the 9 days include travel time or on the ground in NZ?
For 9 days I would skip driving AKL-WLG. Spend time in AKL and to the north and fly to Queenstown. It will take a longer time to drive than you think.
Most rental car require you to drop of in WLG and pick up a new car in Picton after arrving by ferry. The ferrry trip is good
May June will be refreshing, especially in Queenstown
Getting to skiing time.