It depends on how literally you want to translate 'great', we use that title in English as a direct translation of sources written long after his death, the earliest I could find was the 'Historiae Alexandri magni.' Literally 'the history of Great Alexander' by the 1st c. Roman historian Quintus Curtius.

But in his own life time after conquering Persia he himself adopted the title which the Persian Emperors had bestowed upon themselves, 'ShahanShah' (King of Kings) in some later Greco-Roman sources 'Shanahshah' is translated as 'Great King' rather than 'King of Kings' (compare Cyrus the Great or Cyrus Shahanshah')

So, depending on how pedantic you want to be with the translating the answer could be during his life time or as far in the future from it as the 1st c. A.D. Quite a range.

Incidentally, he's not universally known as 'the Great' in Russian for example he's still 'Александр Македонский Alexander of Macedon.

No, the first known occurance of Alexander as the great came from the latin comedy "Mosterallia" of Plautus at the verse 775: "Alexandrum magnumatque Agathoclem aiunt maxumas duo res gessisse"

The fact that this is latin comedy and not some kind of hellenistic history/epic probably means that he was aleady well known as the great at the time (late II century BC). Indeed the seleucid Antiochus added "the great" to his name after his campaign in the east up to the Indus. Moreover the inscription of Adulis (known by a copy made in the VIth century AD) in today Erythrea says that Ptolemy III was also a great king, so it is perhaps even earlier that Alexander was "the great"