If you're flying to MEL to connect to EK404 (DXB-SIN-MEL) then your itinerary is NOT the one you've stated above.

EK does offer a chauffeur service in Sydney - the only exclusion that I am aware of is that it is not offered to people travelling between SYD and CHC/AKL only.

It sounds like you're flying DXB-SIN-SYD; stopover; MEL-SIN-DXB. As you say, you would get the chauffeur on arrival in Sydney. However, as you are not flying back from Sydney, you don't get a second chauffeur in Sydney. You would, instead, get a chauffeur in Melbourne to bring you out to Tullamarine to commence your return journey.

Getting back to Sydney airport for any other further travel you might have is on your own back...even if you have a SYD-MEL flight in there (presumably on Qantas, even with an EK-coded number) all as part of the same ticket.