I don't think they would have thought in that way.

According to Dr. John Walton's book "The Lost World of Genesis One", focusing on the "how" and "when" of creation stories is a modern, Western phenomenon. The Ancient Near East (he includes many creation stories from Mesopotamia and the Bible) focuses more on the "why" of creation, the functional aspects. Therefore, I believe he at least would argue that they would not have been particularly concerned with how old the earth was, in the same way you and I might be. They were probably more concerned with how long things had been working the way they were.

For example, the Sumerian King List opens with the words "After the kingship descended from heaven", and since that is the start of their history, that is what they want to measure, how long their system has been in place, and the functions they observe in daily life between themselves and the deity have been working.

I linked the translation above, and you can see that the number of years ascribed to each king, especially early on, is somewhat unbelieveable, and scholars do not think that each king is actually believed to have lived this long, but if you calculate that the "Antediluvian" kings (before the flood) totaled 241,200 years, and the flood has been dated to around 2900 BC, that gives an approximate date of 244,100 BCE. Further accuracy is difficult, due to overlaps in kings and their reigns in the lists. But that's a ballpark.

Again, I must stress that the Ancient Near Eastern people did not count from the creation of the earth the way we think of it, but rather from the installation of a functioning system, and that was how they viewed creation.