Hi there, we've had a few threads about short connections in DXB here recently. This is the process:
1. Enter terminal on arrivals/connections level, follow red signs for Connections
2. Got through transit security (remove belt, remove shoes if they have metal or are likely to beep, if you're confident they won't you can keep them on)
3. Go up to the departure concourse, find your departing gate
That's it. It's quite simple, basically you just clear security. Now the complication:
Terminal 3 is not really useful when describing DXB airside, it makes more sense to refer to Concourses A, B and C, all of which are used by EK and part of T3.
The walk from A > C can be an easy 20-25 minutes. If you arrive at a remote stand the bus ride from there to the terminal can be up to 30 minutes from touchdown until you arrive. The plus side: EK typically shows connecting gates during the approach of the inbound flight, so monitor the airshow onboard, once you land if you are at a remote stand you might see two buses and staff with signs showing destinations. One bus will go to A, the other to B (A > B requires an underground train, B > C are directly connected end-to-end). IF your destination isn't listed, but you know your connecting gate (B23, A12, C43, etc) ask the staff which bus is going to concourse X (X being A or B, where if your flight has an A gate you want to go to A, for a B or C gate you to go B, as there won't be a bus to C).
It's doable, but it'll be tight if you get a remote stand on arrival, in which case don't waste time at the glittering duty free. Also keep an eye out for staff as you enter the terminal building holding destination signs or calling destinations, good chance there'll be someone for your flight if it is tight, if there isn't someone you've got time.