The Egyptians and Mesopotamians were much more concentrated on the religious and prophetic meaning of astronomy and less on cosmology. We're talking about a very long history and there is no one completely standard model, but very generally the Mesopotamians believed that the flat world was afloat in an infinite ocean; not on the surface, but in the midst. There were waters above the dome of the sky as well. This belief was shared by the Hebrews and scraps of this flat earth cosmology show up in the Old Testament.
The Egyptians believed the earth is flat and oval with a vast Underworld with different levels, gates, and passages. Ra, the sun god, had to make a difficult journey through the Underworld each night. He entered in the west and emerged, reborn, in the east. This cosmology makes some sense if you stay at home and don't travel about. I've never seen an explanation for setting and rising stars, but it makes sense that they too dipped below and rose above the edge after an Underworld journey. I don't know if the Egyptians felt it necessary to even think about that. I do know that they had a special reverence for the circumpolar stars which never set, either during a complete day or the year. They were eternal.
Later, the Greeks had a different, analytical, mindset and were interested in the physical nature of the universe in its own right. Through analysis, and an awareness of what the sky looks like from different parts of the earth, they found evidence for a spherical earth, inside a celestial sphere.
I'll let Aristotle speak for himself:
"Again, our observations of the stars make it evident, not only that the earth is circular [spherical], but also that it is a circle [sphere] of no great size. For quite a small change of position to south or north causes a manifest alteration of [what is visible in the sky at] the horizon. There is much change, I mean, in the stars which are overhead [in the sky], and the stars seen are different, as one moves northward or southward. Indeed there are some stars [e.g.Canopus] seen in Egypt and in the neighborhood of Cyprus which are not seen in the northerly regions; and [circumpolar] stars, which in the north are never beyond the range of observation, in those regions [Egypt] rise and set. All of which goes to show not only that the earth is circular [spherical] in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be quickly apparent."
Simple geometry; which should be intuitively obvious.
Aristotle was just one of many. He was speaking of a standard model that was emerging. One of the arguments for this model was that if the earth were flat, the entire earth would enter night at the same time. They believed it didn't. But this was circular logic, because they had no empirical evidence for "time zones." They weren't perfect.
In the Egyptian model the sunset intuitively looks "right." The sun is falling below an edge. The 19th century Zetetics couldn't ignore time zones and had to modify the more logical Egyptian model, and posited a sun making a circle above the the FE. This makes no analytical sense, but neither does it make intuitive sense. That's not what the sun looks like it's doing. It's an example of how the modern Flat Earthers have to concoct elaborate theories that violate their own philosophy of trust your own eyes.
At first, 21st century FE Believers simply ignored the two celestial poles; (mostly because they didn't know about them). Now they can't. They are now engaged in concocting more and more elaborate theories to explain the two celestial poles. Some of them involve refraction through the dome, some with reflections in the dome of two counter-rotating disks, others more mystical, and some that rely on simple confusion. The modern heliocentric model is much more intuitive, it looks as if we are inside a celestial sphere. The FE Believers are once again violating their own philosophy.