The various cultures of the Ancient Near East spoke a wide array of languages and we know that there was plenty of communication between cultures. We even have a language like Akkadian that served as a lingua franca between many different cultures. Diplomats, and others, must have received training in foreign languages. Do we have any written grammars from the Ancient Near East used for such teaching? Dictionaries? Explanations of points of grammar? Even exercises used by students?
Well, written language was, at the time, an economic tool primarily. It was used to record business, political and liturgical transactions, and to cary on a conversation at a distance through correspondance. The things we use it for, instructive texts (such as language instruction courses) and recreational reading, developed much, much later.
But! There are ancient documents that are meant to instruct scribes in how to learn and teach writing itself - one such is detailed in this online book. (Begins on page 181 "How did they learn Cuneiform?" by Niek Veldhuis). These were word lists, lists of kings, and snippets of seemingly unrelated text, meant as a "primer" for the neophyte scribe to copy.