Before we try to answer any questions about how they accomplished it - we really should ask whether they actually DID accomplish it.
When I google “Sumerian tablet solar system” - I get the image on the right - and a few of the modern interpretations of that (as shown on the left).
What do ACTUALLY we have here?
Well, we have a star-shaped thing with some dots around it.
The person who “interpreted” it did some very dubious things. The bump next to the “Earth” is a tiny soft thing - is that really the moon? Why did they label the biggest circle “Saturn” when Jupiter is by far the largest planet? Why is Mercury larger than Mars?
The “interpretation” has the planets in entirely the wrong order - with no sense that the correct distances were known - and it contains “Planet X” which we know DOES NOT EXIST…despite the claims of the same kinds of crazy people who interpret Sumerian tablets stupidly and repeatedly tell us that Planet X is going to crash into Earth any day now and therefore you should buy their book about it.
But then not all people “interpreting” this agree:
This interpretation has the planets assigned to entirely different dots in the image!
This interpretation isn’t saying “What does this tablet tell us?” it’s saying “I guess that the Sumerians knew all this stuff and how can I fit it into the actual facts?”
Both of these interpretations are utter bullshit…guesswork.
The TRUTH is - we have a tablet that shows what MAY be the sun - and some different sized dots around it that cannot be an informed view of what planets there are.
If this is really is the Sumerian view of the solar system - it’s wildly incorrect.
WHAT DID THEY KNOW?
We know this: Like all people from early human history - they were every bit as intelligent and observant as we are. Furthermore, with much less light pollution they had a vastly better view of the night sky than most of us do - and with no TV or other entertainment - they’d have observed the night sky more carefully and had a better knowledge of it than us.
We know that many ancient cultures were fully aware of the unusual behavior of the planets compared to the distant stars. Mercury, Venus, Mars, Jupiter, Saturn and perhaps Uranus were naked eye objects and they’d have noticed their patterns as they moved across the sky from day to day and year to year.
But without telescopes, they couldn’t have known about Saturns rings - or even that Jupiter is larger than (say) Venus - which seems much brighter.
CONCLUSION:
There is no doubt that the premise of this question is wrong. The tablet isn’t any kind of accurate representation of the solar system. It can’t even be depicting the known planets because there are far too many of them.
What we’re looking at is a depiction of sun and some randomly placed dots representing stars or planets in a non-specific way.The question of how they knew this is moot. They didn’t know it.