Was it common knowledge around 600 AD or perhaps 30 BCE that 200 solar years for example were equal to 209 lunar years? In which period of history precisely can we say that such a thing was confirmed?
Hipparchus, who lived in the 2nd century BCE, calculated the length of the synodic lunar month as 29.53059 days, and that of the tropical year as 365.24667 days (though he expressed both values not in decimal notation but in sexagesimal notation). These values were adopted by Ptolemy and became “common knowledge” among people who were interested in astronomy, whether in the Greek world, or in the Islamic world, or even in Western Europe. The Jews used these values as the basis for the calculation of their luni-solar calendar.
Such things were never "common knowledge" as they are not now. People who dealt with such things are called astronomers, and there were not many of them in any period. Precise relations of the sort you mention were first discovered in the ancient Babylonia, and they were known to the Greek astronomers of the Hellenistic period (probably this knowledge penetrated from Babylonia). (Of course similar relations, but less precise were also discovered in other civilizations: everyone knows indeed that there are approximately 12 months in a year.)
The period you mention (600 a.d.) was the "Dark Age" in Europe (and in the Middle East) when this knowledge was forgotten. There were no known astronomers, and in many places this occupation was even prohibited. Still, the work of Ptolemy somehow survived to our day, which means that it was at least copied. And perhaps those who copied understood something of it. Most of the other Greek works on astronomy were not sufficiently copied in this period, and they are lost. Babylonian astronomy is known from the clay tables which are fortunately durable enough. They were discovered and read in 19 century.