It would be hard for Ethiopia to project power all the way to Egypt at this point in time. It lacks the state institutions to control such a large empire at this time. It possesses no technological, organizational, or morale superiority over the Byzantines. And it is much harder to project and supply an army from Ethiopia across the sudan and Sahara to Egypt than Byzantium could over the Mediterranean Sea.
If Egypt ever rebels though, then it might be able to count on some level of Aksumite support because the Egyptian Monophysite Church is the head church for them.
Aksum might be able to turn the Christian states in Nubia (Alodia, Makuria, and Nobatia) into tributaries, then vassals, and eventually incorporate them into a larger Aksumite state. That process may take centuries though to successful incorporate them for the long term (as opposed to quick conquest that eventually fails).
I think Aksum would have its hands full simply keeping Yemen and expanding into nearby regions of Africa for the next thousand years or so. But control over the Red Sea trade and continued contact with a fellow Christian great power will probably make Aksum and its successor states more developed than IOTL when it was an isolated and besieged power.