Others here have provided some good lists for you to consider, OP. So I'd like to address an important issue in your post, namely, that "accurately reproduce" is a gradient scale. What you are essentially asking about is the history of musical notation (here is a link to a collection of posts on the subject). But the thing about notation is that it always only captures some relevant information that performers find useful, while leaving the "obvious" stuff out. And so now, when that stuff is no longer obvious to us, that missing information becomes just that, missing, perhaps completely irrecoverable.

So the question is what is the threshold. At what point does the information we've lost not matter to your sense of being "an accurate reproduction." This is a question whose answer should have soft, not hard boundaries.

One user below mentioned the Seikilos epitaph. More information about that piece, as well as some additional musical notation that predates it, may be found in this post by /u/KennyWhee. Through a lot of careful reconstruction, we can be reasonably sure of the basic pitch structure of the melody (though some debate is certainly still raging). That's good! But is that enough? It's much harder to know if and how it might have been accompanied by instruments (what kind of ensembles, what kind of textures did they play?). Did they sing it straight through, or was there heavy vocal improv that might have radically transformed the piece in performance? These are things that, in the first instance, the notation simply wasn't set up to handle (noting the complex interaction among different musical parts was a much later development), and in the second, is simply outside of the purview of what the notation may have been expected to even do (even today, singers in, say, the Shape Note tradition naturally inflect their singing in idiomatic ways that the notation does not capture, and when asked, these singers believe they are just singing it exactly as notated!).

Imagine if, 2000 years from now, someone found the vocal notation of Prince's "Purple Rain." This vocal notation might just give a single musical approximation of the vocal melody of the verses with the words of the multiple verses written beneath it (a very common way of notating strophic music), and then of course both the words and music of the chorus. Would such a person be able to "accurately reproduce" Prince's music? In an important sense, yes, at least as well as any of the thousand a capella YouTube covers you can find of the song. But in another important respect, if they can't recover the guitars and 80s synths, if they never hear Prince edging into more strained and embellished singing as the song reaches its climax, and if they can't experience the guitar solo, then there are vast swaths of Purple Rain that this poor future soul will never have the privilege to experience as we do. In that sense, then no, they won't be able to accurately reproduce Purple Rain.

These problems plague any historical enquiry. We will never truly be able to exactly reproduce what music of the past sounded like. We can only get "close enough." But one person's "close enough" is another's "not even remotely close enough!"

Is not exactly the answer to your initial question but we know about medieval music.

here is a great answer by u/hhawks12 that gives an long list of medieval songs with authors, links to the songs and many references. Some of those songs date to the 12 century.

As a follow up question to this, we all "know" what music goes with anything Roman, Hollywood has given us nearly a century of strong trumpeting as a base for anything else Roman, but how accurate is that?