I'm trying to understand one specific aspect of the ol' "how-do-we-know-Alexander-existed-if-we-don't-know-Jesus-did" chestnut.
Just comparing the writings commonly adduced for the two - Arrian's "Anabasis" and the New Testament - Christian apologists often refer to the embarrassment of riches relating to Jesus, in the form of manuscripts for the New Testament, several of them having been dated to within 200 years of their subject's purported lifetime. They compare this unfavourably with the paltry number of sources for Alexander, the best of which is Arrian's "Anabasis", composed nearly half a millennium later and only surviving in a manuscript of more than a thousand years after that. How can we accept the existence of Alexander on such a relative dearth of evidence compared with that of Jesus?
Quite apart from the other issues involved, I'd like to check my understanding of a particular issue relevant to this question. Is it true that our knowledge of Alexander is contemporary, if at second hand, through Arrian, who extensively quotes several of Alexander's contemporaries? I take it we can be sure that these references are real and not made up? Does the Arrian/Alexander situation thus parallel, not the New Testament/Jesus one, but rather, say, our knowledge of what the early critics of Christianity were saying, by virtue of the fact that their words were extensively quoted by contemporary Christian apologists, the pagans' original writings having been lost?
We have extant speeches from contemporary Athenian orators such as Demosthenes that treat Philip and Alexander. We have plenty of Alexander's coinage. We have the claim that eye-witnesses like Nearchus and Ptolemy wrote histories of the campaign, and no reason to doubt those claims any more than any other references other authors make (although there are weird things sometimes - Cicero in his de re publica says he's using Cato the Elder, but it doesn't look anything like Cato's Origines.... I'm working on that problem). We have Plutarch's life in addition to Arrian's history. I think the contemporary coinage and extant oratory would be sufficient though; you don't need Arrian to have evidence of Alexander.
The critique of the New Testament you refer to has to do with the supposed absence of contemporary accounts like the Greek oratory referring to Alexander. Paul is (I understand) our earliest source, a generation or so after Jesus' life. I have heard the argument that the Gospels are the result of tradition preserved orally for several decades; YMMV depending on how much trust you put in oral tradition.
There are some baffling things. Luke for instance claims Augustus made a census of the whole world. Augustus reports several censuses he took in his res gestae, and doesn't mention this one at all, which would have been a massive logistical undertaking and I at least expect he would have mentioned it, since he mentioned several censuses he took of Italy. I don't need to go through all of the supposed weirdness in the Gospels; plenty of other people have done that.
In either case, aside from the Alexander coins, we're using copies of copies - I don't think any autographs survive, so if you really want to be skeptical of Alexander you can throw out the texts I suppose, though we're pretty good at textual criticism and have found papyri that corroborates some of our conjectures. The coins, though, those are harder to ignore.
I heard a debate about a similar issue today, in which the apologist argued there was no more contemporary evidence for Socrates than there was for Christ. I do not wish to argue that Christ was not an historical character, but I will point out Aristophanes was a contemporary of Socrates and wrote a play about him - or seems to have done so. We have the text of the play and Plato felt like it needed a response in his apology whether it is an accurate report of Socrates' trial or not.
Alexander is better off for a) having physical objects like coins extant, b) contemporary writing dealing directly with him and his father, and c) known lost written eyewitness accounts. Throwing out Alexander is like throwing out Cicero's correspondence as though an elaborate work of historical fiction.
Again, I do not mean to be attacking Christianity. The answer is Arrian claims to cite sources. To the extent we trust anything from antiquity, that's the best we get, and sometimes we get quite a lot from such quotations. I think we have some 80 pages of Cato in the Bude edition only from quotations like this.