Does anyone know if any ancient cultures knew the correct structure of the universe? Specifically the solar system and galaxy.?

There is a lot of misunderstanding and urban legends going around. Here are a few I have collected during the years:

The horoscope is attributed to the ancient Babylonians who observed the sky from their ziggurat towers. Well, if we still divide the circle into 360 degrees, today, it is because the Babylonians thought that the year was 360 days (365.25 in reality). It means that nobody bothered to count the number of days because if they did, after a few years, they would be off in the seasons and notice it. How could they then know the star positions when someone was born?

Prior to the Mayan calendar doomsday prophecy a couple of years ago, it was said that the Mayan calendar was more accurate than ours. How can it be? If I say that the time is quart past ten and you say that it is twenty past ten, who is right? If you have only two calendars, how do you know which is more accurate than the other?
It turns out that it was true! The Mayan calendar, when we discovered the Mayans in the 16th century, was more accurate than the Julian calendar we were using at the time but today's Gregorian calendar is more accurate than the Mayan one!

Before, we believed the earth was flat. Well ... no. Any seafarer or anyone with a bit of knowledge knew that the earth was a sphere. But we like to think that our ancestors were stupid.

There was no way to know that galaxies existed in the past. Not until Hubble discovered in the 1930s that what we thought were stars were in fact galaxies of billion of stars. Our own galaxy, the Milky Way, was observed only as that: a concentration of stars in a line in the sky. Nobody thought it could be a galaxy of million of stars around a center.

The heliocentricity has been proposed by some from the antic Greece but wasn't made common before Copernicus. It explained the bizarre orbit of the two inner planets.

And when Halley and Newton predicted that the comet they observed would come back 76 years later and, it did, the world rejoiced, comets being the last of the mysterious heavenly bodies. From now on, everything could be predicted when the "theory of everything" could be found.
... Alas quantum and relativity told us later that the universe is a much more fuzzy place than we thought!

Not precisely but there are some very accurate ideas to be found principally among the Ancient Greek astronomers and philosophers. These ideas are a combination of the very naive with a few gems of incredibly intuitive thinking.

Most early cultures had mythologies concerning the origins and structures of the Universe. There are a lot of attempts to compare these to modern cosmology, but this is stretching it. Even if you look for hints of Relativity in the Ancient Greeks, as has been done with Heraclitus, you certainly cannot say that Heraclitus understood relativity or invented it! Sometimes these profoundly intuitive men did arrive at incredibly prescient conclusions.

Democritus believed the Universe was composed of infinitesimally small and indivisible particles he called Atoms. He wasn't entirely correct because even Atoms break up into smaller particles. But who can say Democritus truly anticipated modern physics? Electricity was not at all understood.

Democritus is also credited with suggesting the Milky Way was composed of tiny unresolved stars. Anaxagoras also is credited with the suggestion.

But none had a concept of galaxies... that our Milky Way is but one of a myriad of "Island Universes." That had to wait until Immanuel Kant and the Edwin Hubble before they were truly understood (the Herschels and Lord Rosse opened the door on this).

Scientific reasoning had nothing to do with many of the early cosmologies- these were based on vague mystical beliefs. People love the thought that some Hindu or Buddhist cosmologies, or the Qur'an, correctly surmised the Big Bang Theory or other tenets of modern physics an astronomy (even quantum mechanics!) but these ideas easily collapse under scrutiny.

No, generally the more advanced cultures had the fact that the Earth was a sphere and they could predictthe path of the planets and eclipses. Beyond that they thought that the Earth was the centre.

There aren't any