So where does Cyrus the Great fit in? I’ve always been told he was the golden boy of Persia.
The first source to refer to Alexander as "the Great" was long after his death, I believe. In his lifetime he would be called "Alexander the son of Philip" if "Alexander" by itself wasn't sufficient.
He was not popular with most Greeks of the day--Darius III had large numbers of Greek mercenaries fighting on his side. After Alexander's victory at the River Granicus, he had the Greek mercenaries he captured slaughtered in keeping with his myth that he was fighting a war of revenge (because of Xerxes' invasion--Macedonia had been on the Persian side in that war).