1st, the title of your query is profoundly misleading......Uber, taxis, Apex, Jucy etc. are not public transport.
As you have already discerned, public transport in Melbourne & Sydney is excellent. Better than comparable sized American (& perhaps Cdn.) cities!
New Zealand is another story. Public transport is available, but is very weak. I relied on public transport in Dunedin, for example, and spent more time waiting than riding.
Queenstown receives large # of int'l visitors and does have public transport buses and a 7-day visitor card:
http://www.connectabus.com/