We are currently in Sydney having breakfast overlooking the opera house. will be taking BA16 back to SIN this week.
We have done this trip many times and my first advice would be break the trip in Singapore. We have done this trip many many times in one flight and even though we always fly in F, you feel much better by breaking the trip in Singapore for a night. Breaking the trip in Singapore also allows you to fly the A380 to SIN. Even though you are flying Club you will be very keen to get off the plane in Sydney.
Try to keep yourself hydrated on the flight, the air quality on the 777 is very dry. Try to eat smaller portions as flying this distance disrupts the digestion.
You need to try to sleep on the SIN SYD sector as the flight arrives at such an inconvenient time particularly if you are checking into a hotel. We were renting a car so picked the car up drove to Manly and had a great breakfast and walk before heading back to the hotel to checkin.
It will take you up to a week to get into the time zone in Australia.
On the way back, try to keep awake on the SYD SIN sector and then sleep on the SIN LHR sector. Again it will take you some time to get back into UK time.