Also slippers while showering would be a good idea, I haven't personally seen the SYD public showers but based on other public showers don't expect the cleanest. As stated above bring your own soap/shampoo and towel. Or buy some form of AA pass to the AC which will get you access to the Qantas Club lounge at SYD domestic terminal airside.(note AC day pass no longer gets access to partner airline lounges, so the cheap $50 option is out).