Welcome to FlyerTalk. It's good you came and asked, as you do not have a lot of time for these transfers! You'll certainly have no time to dawdle or shop.
When you pass through immigration and exit customs and walk (fast) to your right, you'll spot the proper Transfer Lounge close by the McDonalds sign. You'll process security before accessing the lounge. "Qantas Seamless Transfer" (if only!) busses depart as short as 10 minutes headway, as long as 30 minutes. The bus travels across the airport via a roadway within the airport itself via the western runway threshold and arrives Terminal 3 at Gate 15.
T-3 Gate 15 is where you will pick up your bus back to T-1, where you'll have to process immigration and security yet again before trekking through Duty Free toward your gate (or the airport lounges if you have access). Frankly, I'd recommend trying to get an earlier flight from MEL - SYD if you can (assuming there even is an earlier flight); the idea of one hour for transfer T-3 to T-1 makes me cringe. In any case, you best bet:
Qantas Seamless Transfer
If you are transferring between a Qantas domestic service (QF400 - QF1599) and a Qantas international service (QF001 - QF399) in either direction, you may be eligible to use the Qantas Seamless Transfer Service. For more information and to confirm your eligibility, visit
www.qantas.com or phone 13 13 13.
See
here for all SYD inter terminal transfer options.
For a PDF map with all options, click
here
Note: I added some to the thread title for easier / broader search by other members seeking this information. I've redacted much of what was posted here and created a Wikipost for others with similar questions. /JDiver, Senior Moderator