Before the discovery of magnetism, there's no particular reason to have north be "up" on a map. North is up because magnets point that way, so if you align your compass with the compass arrow on the map, the things on the map will be aligned with the things on the ground. That's called "orienting" yourself.

But wait! Orient also means "the east", which is why Europeans labeled people from Asia "Oriental". There's a reason that the two words are the same. Before magnetism, you needed some way to make sure the map and direction you are facing agree with each other. So how can you be sure what direction you're facing, if compasses hadn't been invented yet? Well, every morning the sun rises in the East. So at dawn, you get out your map, look toward the rising sun, look down at your map, and everything should align properly.

Now, not all pre-compass maps had east at the top. There was no real standardization. But east was most common, which led to "orienting" yourself, and other words like "orienteering" as finding your way, and "orientation" as being shown around.

It should also be noted that Christian doctrine also said that Eden was to the East (of Europe, at least), providing another rationale for so many maps with east at the top.

Here are some pre-compass maps. They can be hard to decipher, because the drawings are not nearly as accurate as our modern maps are.

This one is a sort of idealized map of the world with no detail, but you can see that Asia is above Europe and Africa.

This is the same sort of map, but with a lot more detail. That big blue thing that stretches up from the bottom? That's the Mediterranean. Because it kind of looks like a T, and the whole map is circled by an ocean, both of these are known as T-O style maps.

This is an Islamic map of the world, with the south at the top.

This is another south-up Islamic map of the world. It's actually much better than the others I've shown. Here you can see it North up, and see how recognizable places are, compared to some of the stuff above.

I don't know off the top of my head any maps with west at the top, though I would not be surprised to find some. And I'm not familiar with Asian, African, or pre-contact American attempts at world maps. There was an awful lot of trade from China and India and the Horn of Africa through the Indian ocean, so there are probably some interesting maps of that area too.

I have no source to back this up but I vaguely remember listening to something by an astronomer who said that the first photos of earth that were taken showed earth "upside down" but the Western world wanted to depict themselves as the ones above everyone else so they turned it. Of course we're just a sphere in space and there's no right way the earth faces.