The oldest known writing we know about is cuneiform in clay tablets. Some tablets describe how to get the intervals and notes, and then those are used by name (as in play this and that).

The second part of the question is very complicated. There's the theory part, and then there's the performance practice part.

We can follow instructions and get the intervals, but will we have the exact same notes? Probably not, and that was probably not a problem back then (but who knows for sure). In the last centuries, in the West, we have worried about being rather specific when it comes to notation and what not, but people were playing quite different sounds for the same note name. Getting the relationships between the notes is one thing, and getting the exact same notes is another (something that has not been much of a problem).

Now, the performance practice part makes it worse. You can have a model of how to tune the notes, and you can even have a tuning standard like we do, but what if somebody wants to bend a bit here and there? Notation has not been able to convey a lot of information to perform the music.

Hi, hopefully one of the music historians will drop by, but in the meantime, you can get started with a few previous posts on the subject

• When did music begin to be written like it was now and how did people write it before? did it vary from place to place? featuring /u/flotiste

• how did the standardized musical notation come about? featuring /u/catsandtravel, and also follow the link at the bottom for more threads

A couple of other posts I ran into while searching, which you may find interesting

• /u/erus shows some examples in post So is the typical classical score we see today the only standard notation of music or were there other civilizations prior who had their own?

• What do we know about Roman music? featuring /u/casestudyhouse22