Yes, if the registration is open at that time and if you are allowed to go against the grain and walk back airside in T1. (If you try this, please let us know if it works.) But I am not sure the point of using the T1 option -- wouldn't it be faster just to go through immigration in T3, use the desk there (if it's open), and go back through immigration to catch your connection? You should be able to then use the e-gate to leave Dubai if they issue it right away, further saving time.

Given the generally shorter immigration queues in T3 compared to T1, and the long walking (usually; unless you are lucky and your EK flight arrives at T1) required airside from T3 to the T1 exit, I don't think that this would save time anyway. But you'd get plenty of exercise.