From Sydney to Wellington, Air New Zealand only use A320's which are fitted with all economy seats. There is a seperate product at the front of the aircraft (first 2-3 rows) called Works Deluxe but this is the same service as economy with just a spare seat between you plus priority check in, boarding, a bit more legroom and slightly better headphones for the IFE. A works deluxe upgrade can not be done with any points only cash and typically is only $130-200 depending on the route.
So ignoring all the above it is possible to upgrade Star Alliance flight using points however it's basically impossible. Here are the steps:
1. You need to be on a flight that has a business class cabin, from Sydney this is Sydney to Auckland only (noting that only 787 and 777 flights are eligible). Typically only one flight per day is operated by an aircraft with a business class cabin.
2. You need to be booked in an eligible economy booking class this is U, E, O, Y, B (oddly some of those fare classes are Premium Economy but they're not eligible for upgrade). These booking classes are the most expensive flexible fares the cheapest
oneway SYD to AKL flight in one of those economy fare types is $878.
3. There needs to be business class award space on the flight you are looking to upgrade. Air NZ award space in general is limited and in premium cabins basically non existent (except for a handful mentioned in this thread).
4. Once you have an eligible booking and there is award space you need to put in a request to Singapore airlines. The upgrade cost is 23,000 krisflyer miles which if you converted VA points would be 35,650 points for the upgrade. The requests are done via the phone which takes time.
In total you'd be spending at least $878 plus 23,000 krisflyer miles assuming there is even award space in the first place.
If there is award space you could book the business class award for 34,500 miles plus ~$120 taxes.
It's also worth noting that paying $878 for a fully flexible economy fare just to upgrade is terrible value when Sydney to Auckland is currently on sale in business class for ~$500 oneway.
You shouldn't need the points in your account to search. Also you can search Star Alliance award flights via the United website without logging in or via other websites eg :
Star alliance award search - Travel hacks