DMPCHEAPIESNZQ2
The old code has expired at the end of March, and Uber sent me a new code yesterday. Use coupon code DMPCHEAPIESNZQ2 when you sign up to Uber New Zealand to get $20 off the first ride. The new expiry is 30 June.
Here's a disclaimer: this is in fact an affiliate code for ChoiceCheapies, and by using this code when you sign up to Uber & take the first ride, ChoiceCheapies will receive commission from Uber. ChoiceCheapies' policy has been not showing affiliate links to logged in users (and guest users can opt-out), but in this case we are unable to not receive commission when the code is used. Alternatively you can also use this code from Groupon to get $20 off first ride if you are not comfortable of ChoiceCheapies getting affiliate commission. Or use the referral to split the $20 with the referrer.
Note: I've basically posted the same thing on OzBargain.