This is a tough question to answer because we do not have records from those days. However, the name certainly existed and was certainly used often. According to John Chadwick's The Mycenaean World, the first example of the word was in Linear B as (Alexandria), the feminine version of Alexander. The name was also used as a title or epithet given to those who do good deeds in battle. The name itself means "defender of men" and is usually applied to warfare. For example, if a soldier was to save a few of his soldiers by defeating the enemy in battle, he may be given the title as recognition. King Priam of Troy had two children who were given this name, Alexander and Alexandra. We know them better as Cassandra and Paris, but they are also known by these names.

To answer your question, we have no way of knowing how popular the name was prior to Alexander III of Macedon. However, we can confirm that it was used prior to his birth and by those in power. Sadly, the only names we can track in ancient and classical times are those who were worth mentioning. Obviously, there were two Alexander's of Macedon before Alexander III.

Chadwick, John. The Mycenaean World. New York: Cambridge University Press, (1999).