The Sun still only illuminates part of the disk at a time. It doesn't go below the disk at night, it's still above the disk just too far away to see, so you get different times of day in different parts of the world.
Yes, this just raises more questions. Yes they have answers for them. None of them are good and very few of them are even internally consistent, let alone hold up to any scrutiny.
I used to work with one. The notion is that the sun travels in essentially a circle above the earth-disk and is also much smaller and much closer than you've been led to believe. They believe that the world actually is exactly as it looks in an azimuthal equidistant projection centered on the North Pole, and that it being what the world really looks like is why that's the map on the UN logo. Antarctica is essentially the rim of the world and what keeps the ocean from pouring over the edge of the disk, like the rim of a giant bowl.
You dig deep enough and you learn that they also don't believe in gravity (because if gravity was real then it would tend to pull people nearer the edge of the disk at a deeper slant relative to the surface). It's just that the Earth-disk is accelerating upwards through the void at 9.8 m/s^2.
People assume that time is a strict progression of cause to effect, but actually, from a non-linear, non-subjective viewpoint, it’s more like a big ball of wibbly-wobbly, timey-wimey stuff.
Most people think time is like a river that flows swift and sure in one direction, but I have seen the face of time and I can tell you, they are wrong. Time is an ocean in a storm.
Even without time zones, 5am is an hour from the end of my work night. I'm absolutely sitting around on my phone at that point. I just looked at the clock and it's 5:02am, so it legit would have been 5 on the dot when I saw this.