Pompey certainly earned the comparison to Alexander, especially with the Pirate Command, but the title "the Great" popped up very early and in a not exactly flattering way.

When Sulla returned from his war against Mithridates in the East (82 BCE), he marched from Brundisium north to Rome, displacing the generals of (the now dead) Marius' party and gathering his own supporters along the way. Among these supporters was a young Pompey who raised troops himself and put them at Sulla's service. Sulla sent Pompey first to Sicily and then Africa to displace Marian forces in the provinces (exterminating the Marian party would take until 72 - Sertorius was a slippery opponent in Spain and only done away with by conspiracy of his own officers, even though Pompey got the credit). After Pompey dispatched the Marians in Africa, Sulla ordered the return of the army, but not the general, who was to wait for his replacement. Plutarch says Pompey's troops didn't take that well (we can guess who really didn't take it well) and would only return with Pompey. Sulla, according to Plutarch, who had the man's memoirs so is as well informed as anyone could be expected to be, thought Pompey might have been in rebellion himself. Sulla had also had trouble with Marius' nephew Marius Gratidianus, and Sulla was joking about being an old man fighting young mens' wars (the young men being Pompey and the younger Marius). When Pompey returned from Africa with his troops instead of remaining as ordered, Sulla greeted him as "Magnus" "the Great" in what seems to me (I think that's an honest reading of what Plutarch presents) an attempt to diffuse both the possible insurrection and what we might call the "optics" of the situation.

Here's Plutarch on that moment:

Sulla's first tidings of the affair were that Pompey was in revolt, and he told his friends that it was evidently his fate, now that he was an old man, to have his contests with boys. This he said because Marius also, who was quite a young man, had given him very great trouble and involved him in the most extreme perils. But when he learned the truth, and perceived that everybody was sallying forth to welcome Pompey and accompany him home with marks of goodwill, he was eager to outdo them. So he went out and met him, and after giving him the warmest welcome, saluted him in a loud voice as "Magnus," or The Great, and ordered those who were by to give him this surname. Others, however, say that this title was first given him in Africa by the whole army, but received authority and weight when thus confirmed by Sulla. Pompey himself, however, was last of all to use it, and it was only after a long time, when he was sent as pro-consul to Spain against Sertorius, that he began to subscribe himself in his letters and ordinances "Pompeius Magnus"; for the name had become familiar and was no longer invidious.

There are some other interesting comments made at that meeting. Pompey asked for a triumph, which was a shocking request for a fucking 20-something kid that hadn't even held elected office. Sulla said no, and Pompey said something about more people worshipping the rising than the setting sun, and Sulla changed his mind. Tacitus reports that Tiberius and Caligula repeated nearly this conversation with that exact phrase involved, and I'm not sure what to think about that yet, but the moment seems to have had significant cultural weight. Pompey did celebrate his triumph, and had his triumphal chariots drawn by ele-fucking-phants, which didn't fit through the gate of the city and he had to resort to horses. [Profanity is completely appropriate for what Pompey was up to]

Plutarch goes out of his way to point out that "Magnus" was a ROMAN and not GREEK honorific, indicating the precedents of Valerius and Fabius who earned the name "Maximus" and not necessarily for only martial virtues. So not only was the comparison to Alexander first made as something of a joke, but the ancients knew it wasn't fitting when it was made and attempted to justify the title by comparisons to other people.

Many years later Pompey did conquer the East after Rome fought a number of wars with Mithridates of Pontus, and he reorganized eastern political geography, including entering the Holy of Holies in Jerusalem. By the time Caesar crossed the Rubicon in 49 Pompey was by far Rome's most accomplished general since Scipio, and probably deserved to be ranked with Alexander as a great commander.