While nearly the entire final arc of the Yu-Gi-Oh series is dedicated to the memories of the pharaoh it doesn't really give an altogether fantastic picture of the world he lived in. The manga especially just jumps right into the plot without really giving us much exposition at all. For this reason I tend to assume that for all intents and purposes it functioned more or less like your standard run of the mill Ancient Egyptian society with some magic and card games thrown in there. Which really is not too hard to adapt considering Ancient Egypt like many of it's contemporaries held strong religious and mystic beliefs that may as well have been a reality to the society of those times.
So on a scale of one to Ancient Egyptian card games, lets start with the basics. According to Ishizu during battle city, Atem lived during the Eighteenth Dynasty of the New Kingdom and Takahashi himself has connected Memory World with the Amarna Period. This is plausible because many events of the around this time can be paralleled with Yugioh canon. The New Kingdom and the Eighteenth Dynasty in particular was the imperial golden age of this civilization. After the expulsion of of the Hyksos, foreign rulers in the north, Egypt was no longer content with the natural borders of it's land and adopted a great expansionist policy. Basically it was a time of war and conquest. Yet by the time of Amenhotep II and III, they were finally starting to calm down and feel comfortable with the reaches of their empire. During this time the foreign policy of the throne transitioned to one based more in maintaining their diplomatic relations for the sake of prosperity and stability. While the name's don't match up, the reign of Atem's father, Akhenamkhanen, was also plagued by the end of a long war. The millennium items were created to speed the end of this war and ensure Egypt's dominion once and for all. And Atem's reign is going relatively well until a collective of evil forces basically tears it all apart. The important one here is his uncle Akhenaden, who's name is bears a striking resemblance to Akhenaten, successor of Amenhotep III, the heretic king who also led Egypt to a rather unfortunate downfall only to have it set right again by his son Tutankhamun. Similarly it is Set, Akhenaden's son that becomes the next pharaoh and is charged with setting things right again.
So the setting is the Eighteenth Dynasty during a period of unparalleled peace and prosperity, at least until the events that lead to Atem's downfall. But Ancient Egypt was a very conservative society rooted in observance of traditions and the past so many ideas and practices were passed down from the earliest dynasties. By the New Kingdom, society had a wealth of material to draw upon as well as plenty of material that substantiates their reliance on such a storied history.
Obviously Egyptian life was simple by modern standards. Theirs, like so many other ancient civilizations, was an agrarian society and their dependence on the Nile river is well reflected in their development. It was the lifeline of their society, reflected in their own view of the world. Egypt was called Kemet in their language which means black land, the the color of the rich soil irrigated by the river. As opposed to Desheret, the red land of the desert for which they felt little affinity. The Nile was also a special case. Unlike the Tigris and Euphrates of nearby Mesopotamia, the Nile flowed regularly, flooded nearly without fail every year and it served as a dependable source of stability that played a key role in allowing Egypt to become such a dominant force in the Ancient world, marked by long periods of a strong central bureaucratic government. Conversely Mesopotamia was rarely more than a collection of city states due to the erratic nature of the rivers.
This stability allowed Egypt to remain deceptively simple. In technology and many of the sciences Egypt actually lagged behind her counter parts. Regularity, like that of the inundation, defined the life of the average Ancient Egyptian. This meant that they rarely needed to change their ways or develop new methods to cope with their problems because they arose so rarely. It has also been said that once they found an acceptable solution to a problem they rarely revisited it which meant that their innovations developed only so far as their needs required and that made them an altogether practical culture as well.
Society itself was divided into pretty regular classes. The largest group is of course the poorest, made up of the farming peasants as well as servants and slaves though the idea of slavery in Ancient Egypt is probably different than it is understood in a more modern context because Egypt had no monetary system. The people throughout the society subsisted on bartering and their pay for services consisted of room and board basically. Depending on how important they were or how well they did their jobs they might be gifted more than the bare necessities as well. Again, Egypt is unique in the fact that, while there is always the possibility for abuse, especially of this class it was probably less necessary in Egypt due to the fact that the land was so rich and well organized that the state could in fact provide for the vast majority. It is also notable that the state and the king, while not directly involved in the affairs of the low class people, were intimately aware of the important role they played in maintaining the state. Individuals may not have been viewed as such but they could still be treated well enough to ensure that they do their part to provide for the country which would surely fall to pieces without such important work.
Egypt also saw the rise of a rather substantial middle class. These were made up of independent workers and traders as well as the special craftsman and artisans which were a luxury and a product of the rich society. In the New Kingdom in particular the state supported an entire village, known today as Deir el Medina, dedicated solely to working on the tombs of royalty which is just another example of just how rich they were.
The most well known class and the most relevant in this case was that of the upper class, the nobility and royalty. The former was made up of viziers and other important officials who formed the governing body apart from the pharaoh. In the world of Yu-Gi-Oh this role was summarily assumed by the priests although in reality theological institutions had very little political power. But other than their title and role in society, their function in court appears to be more or less they same. They are the judges with Pharaoh as their head. Not that they would be expected to deal with all problems of the country. Only matters of the upmost importance, such as those that concerned the security of the state as well as those that had excalated from below came before the great court at Thebes. Egypt is a very long narrow piece of habitable land and while the river aided transportation greatly it was still a complete chore and people general prefered to stay in one place for long periods of time if it could be helped. Therefore power was delegated throughout the country. This does not mean the power of the pharaoh was in any way diminished. The pharaoh posessed absolute power in all matters of government, economy and religion and it was well understood that the people he appointed to the various positions were simply fulfilling the multiple roles required of him on his behalf and they were all subject to his review at any time.
References:
Pharaoh's People: Scenes from Life in Imperial Egypt By T. G. H. James
Ancient Egypt: A Social History By B. G. Trigger, B. J. Kemp, D. O'Connor and A. B. Lloyd