There is one practically at the airport - the North South. Worth considering as hiring a 'van takes longer than you think, then there's stocking up, access to somewhere to eat if you can't face cooking after the flight, the tiredness from the flight itself... The only one that matches what you are asking is Glentunnel - just outside Darfield.
Honestly, please don't drive anywhere if arriving on a long haul flight. Yes, others have tried, and a sad number have failed. Driving jet lagged and sleep deprived is a killer.
Spend day 1 exploring Chch on foot - natural light and gentle exercise is the best way to combat jet leg. Have an early dinner, early to bed and the following day collect your camper and head off on your adventures.
Please plan wisely and travel safe.
Ask your campervan provider about the check-in process and how long it is expected to take. It can be quite a long process going through all the dos, donts and how tos after a long flight.
Please consider picking up the van the next day.
Or could consider the Riccarton camping ground - about 10min drive from the airport. There is a big supermarket not far - you can use to stock up before heading off.
Omaka Scout Campground is open to the public, which is very close to the airport. You can easily go around the back of the airport to get onto SH 73 in the morning.