No unfortunately.
There are some public landside showers (but BYO towel soap).
Most Sydney hotels are accustomed to early arrivals and if a room isn't available can generally sort you with gym and shower access.
Shower facilities are available on level 3 of the T1 International terminal, before security. There are also shower facilities located airside (passenger only access) near Gate 31, Gate 51 and Gate 24 (male only) on the Departures level. You will need to provide your own supplies for the use of the shower. These supplies may also be purchased at some of our retail outlets.