When buying the bottle in SIN, you get the duty free store to deliver it to your boarding gate (ie. clear security at the gate and then collect your purchased duty free before heading onto the plane). The duty free allowance for Australia is 2.25L of any alcoholic liquid, so you can buy up to that limit (essentially 3x750mL bottles) without needing to declare anything on arrival or pay any duty or tax in Australia.
At present there are no liquid restrictions on domestic flights, with the exception of domestic flights that depart from international terminals (a couple of examples being QF8 BNE-SYD and JQ35 SYD-MEL). Basically:
If your flight is on Virgin, you're fine.
If your flight is on Qantas, you're fine if the flight number is QF400 or above (QF1-QF399 depart from international terminals).
If you're on Jetstar, you're most likely fine, but just double-check the terminals.
Of course, if you're in any doubt at all, feel free to post any/all of your flight numbers and I'll be happy to double-check the terminals for you.
PS: Obviously, if you were looking to take the alcohol home with you on your final leg to SIN, you aren't able to (unless it's in your checked baggage), as the standard 100mL restrictions apply on international flights ex-AU... in that case, simply buy the alcohol duty-free on arrival at Changi subject to the Singapore duty free limits.
