In this answer u/Trevor_Culley describes how the romans didn't have much understanding of ancient Assyria: https://www.reddit.com/r/AskHistorians/comments/sisgn2/what_did_grecoroman_historians_c_500_bce_to_500/

Based on this Sumer and Akkad seems way before the time horizon you could reasonably expect them to understand, although I'm sure more can be said. As late as 539 BC king Nabonidus at least had some idea of Sargon (though vastly overestimated how long ago he ruled) so that past was, vaguely, still known.