MELi airport should open before check in for the first flight to DXB.
I was there for the EK flight to AKL a while back, the first door to int dep does not open before 5am, you can do OLCI, but would probably still have to do passport swipe.
I stayed at the Ibis Budget that time, was in line in the J premium checkin at 4.15am, but noted the frosted glass doors didn't open till 5am.
So your choice, you could be there at 4.30am, there would be people queing up at the frosted door even at that time.
Granted I haven't passed int sec and doing passport outgoing Smartgate since the security upgrade, but no early than 4am I would say should be ok for a 7.xx flight.
You do need some J lounge time too, no matter how "bad" the dungeon might be.
Edit: I think they lock the frosted doors after that first EK flight from MEL - DXB is processed through, because at 5.15am when I did it, definitely, the frosted door was locked, with people lining up, and the J lounge had a sign it opened at 6am.