You should nail down a time period. Egypt is quite old, and the African cultures or kingdoms rivaling Egypt will change depending on the time period. So, without guidelines, I'll assume you meant Egypt at its height.

Egypt is generally recognized to have three peaks of power and influence; the Old Kingdom period, the Middle Kingdom period, and the New Kingdom period. Those three are sometimes referred to as "Golden Ages".

During the Old Kingdom, there were no civilizations to rival Egypt in Africa. It should be noted that in many parts of sub-Saharan Africa, the Neolithic era proceeds directly into the Iron Age, there was no Bronze Age to speak of. The time of the Old Kingdom is also a time of considerable strife in Africa, there were huge movements of sub-Saharan Africans into Western Africa, possibly because of climate change. You could possibly consider the Nubian civilization to be a great contemporary of the Old Kingdom, but they'll be dominated, conquered, and integrated by the Old Kingdom at various points, and then Nubia is going to turn around and conquer Egypt, but since they use many of the same symbols it becomes very hard to decide where Nubian civilization starts and where Egyptian civilization begins and visa versa, so I'm not sure they'd fit the bill.

During the Middle Kingdom, the only one I can come up with is Punt. One of Egypt's trading partners, it was a rich and powerful land. It produced a lot of valuable goods for the time, like gold, incense, and furs. But, there is some controversy as to where Punt was actually located, some think it was only located on the south of the Arabian peninsula, while others think it was primarily located in present day Somalia, while still more think it covered northern Somalia, Eritrea, parts of Ethiopia and the south-eastern end of the Arabian peninsula. Some of our sources even place Punt in Palestine. But again, Punt is always viewed as falling under Egyptian influence, so it may not fit the bill either.

During the New Kingdom, Egypt will have it's Ramesside period. Giving Egypt the largest extent it will ever have, giving Egypt the most influence it will ever have on its neighbours, and giving us some of the most well known pharaohs. No culture in Africa at the time even comes close to rivaling New Kingdom Egypt. It was a cultural, military, and artistic power-house.

So, to answer your question, no, Egypt during it's heights, never had rivals in Africa. The closest candidate in my opinion is Nubia, but they were so culturally linked with Egypt at various points that it's incredibly hard to separate the two, they are sometimes seen as being two sides of the same cultural coin. Much like we see Greek and Roman paganism as coming close in terms of cultural identity. Aksum rises from the fall of the New Kingdom, and Carthage will come some time after Egypt is an independent, dominant power, and in my mind is the only real contender until we see the rise of Medieval Africa with the likes of Mali under Mansa Musa, the Songhay or Benin empires.

Sources:

The Egyptians, by Alan Gardiner.

The Egyptians, by Cyril Aldred.

Egypt, and the Egyptians, by Brewer and Teeter

Are you specifically asking for cultures that existed at the same time as ancient Egypt, or are you asking about the existence of rich and technologically advanced African cultures in general. Also, what technologies are you interested in specifically?

If you're talking about wealth, obvious choice is mansa musa and the mali empire which if the inflation is correct had a hundred times the wealth of Egypt and most of the world at it's prime, but Carthage is also a notable empire that rivaled not only Egypt but also Rome. Technology wise what are you referring to? if it's warfare any number of kingdoms used similar weaponry as Egypt with chariots, Nubia would later go on to conquer Egypt and replicate their pyramids so on some level i assume they had the same understanding and know how of Egyptian masonry even though their pyramids were smaller. Ancient/medieval Somalia was also adapt at naval trading and warfare, at their height they were a force to be reckoned with but the most notable kingdom i can think of that could be rival to Egypt all around so far as wealth and technology would probably be Ethiopia, like Egypt they were able to expand their territory outside of Africa, both were reasonably wealthy and have interesting archaeological/historical sites. Plus they also had noticeable political clout in other regions outside of military influence.