SQ has no arrival lounge.

If you are already checked in for CHC-SYD (ie it is part of the same ticket as SQ) then you could stay airside in international terminal. However I'm not sure the international lounge will be open all day (not too many international flights from CHC).

Otherwise you could visit the domestic lounge (which means going through immigration on arrival and departure at CHC). If already checked in just show your international boarding pass. If not then you'll need to show your ticket. Check in at CHC will only open about 3 hours before departure.

In either case make sure your checked bags are checked all the way to SYD (can do this even on separate tickets) otherwise you'll need to store them in order to visit the domestic lounge (it is airside and cannot take large bags through xray).

Airports in Australia/NZ often have people checking carry ons for size/weight just before immigration.