A great quote from the book: "The situation we are faced with can be summed up roughly as followes. Inspired by the example of Theopompus and his pioneering Philippica (a history of Greece written around the career of Philip II of Macedon, Alexander's father), more than twenty contemporaries wrote histories or other kinds of work on Alexander. Not one of these histories survives in the original. Of the many letters ascribed to Alexander, just one extract of one of them has a better than average claim to being genuine. The earliest surviving connected narrative account of Alexander's campaigns was composed in the first century BCE, some three hundred years after the events it relates; it is, besides, only incompletely preserved. Thus the sole connected narrative to have survived complete is a third century CE epitome, or abrigment, of a first-century BCE work in Latin by a Romanized Gaul. Finally, what is generally today accounted the best of the more or less completely surviving histories of Alexander was written by a Greek philosopher-statesman in the second century CE, probably during the reigh of the philhellinic emperor Hadrian. In short, 'it is as if the history of Tudor England could only be recovered from Macaulay's essays and the histories of Hume the philospher' (Robin Lane Fox)."
I'll admit, being an American I don't quite grasp the full context of that quote but you can get the gist. Basically, I wouldn't take this letter for actually being true (which the title rightly states as being apocryphal) and rather read take it in its own context of someone, probably a much later Arabic (edit: or Persian) writer using the story of Alexander as a means of conveying their thoughts.