There are public showers in T1. See
here. Have to use your own towel though - none provided. Never used the facilities, but seems like much less hassle than the alternatives.
Originally Posted by Sydney Airport Website
Showers
Complimentary shower facilities are available through the terminal at the following locations. There are currently no towel vending services available.
Departures:
- Northern end - behind Check-in Counter B
- Southern end - behind Check-in Counter K
- Gate 10 - Passengers Only
- Gate 30 - Passengers Only
- Gate 59 - Passengers Only
Arrivals:
- centrally located behind the glass lifts