There is very little information about what he did there in the first place.

Here is pretty much what we know of him in regards to Egypt.

Since the Achaemenid Empire's military functioned on satrap levies, the Satrap of Egypt, Sabaces, was ordered by Darius to levy the troops in his satrapy and lead them to join the Imperial Persian army that was routed at Issus. Sabaces himself was killed. So when Alexander arrived in Egypt, there was no resistance, so we can only assume that only Persian garrison troops remained (Because the Egyptians had only recently revolted against the Persian Empire).

He ordered his fleet stationed in Phoenicia to advance and Egypt and marched there from Gaza, and when he arrived at the mouth of the Nile river, on the Eastern-most branch of the Nile delta, at a city called Pelusium, the place where usually defending Egyptian armies try to defend the country from asian invaders, he found his fleet already moored. There, they were recieved by the new Persian satrap Mazaces, who surrendered to him the royal treasury and parapharnalia.

Alexander organized a garrison in Pelusium and sent his fleet upstream to the satrap capital Memphis, while he didn't cross the Nile, and descended South to Heliopolis, a bit further North than Memphis.

After passing through Heliopolis, he crossed the Nile and arrived at Memphis. There, I imagine he acquainted himself with the local elites and administration, probably with the help of the ex-Persian satrap. And sacrificed to the local gods and threw several festivities for his army and peoples.

After a while, he sailed downstream with part of his army, detoured to the Western-most branch of the river delta, and in the coast, allegedly he ordered the founding of his new Alexandria, specifying the locations of the walls, the Agora and temples.

There, he recieved a visit by Hegelochus - a vice-admiral that had been tasked to fight the Persian garrisons still holding many Aegean islands, as Alexander was still pushing into Anatolia - and brought him the news on his front, with the conquest and submission of several islands.

After which, on a whim, Alexander decided to journey to Siwa Oasis to visit an oracle there, that was said to have been visited by Perseus and Hercules, characters that were respectively largely associated with the Achaemenid Emperors and with Alexander.

He went along the desert coast West to Paraetonium and then turned South towards Siwa Oasis. According to the authors, there were no major logistical issues associated with the journey, other than his army being lost and being shown the way by ravens. It is unknown how many men he took there.

Once there, he visited the Oracle and came back wanting to be called the son of Zeus or of Ammon. It is likewise unknown if he left behind some sort of military presence to hold the oasis, or if he left them to their own devices after recieving their alligeance. Likely this was because he was in a land that deified its rulers, and felt like a way to be accepted by its new subjects and native elites.

He eventually made his way back to Memphis, where he recieved several embassies from Greek cities, along with reinforcements. Then he threw some more festivities, sacrificed some more to Zeus. Curtius says that then Alexander had a large desire to visit Southern Egypt and Kush, and indeed he sent Callisthenes and Aristotle to Kush, possibly on a diplomatic mission. It is likely that he spent some idle time, but eventually he recieved news that Samaria in Palestine revolted, and that was the trigger to depart back to Syria.

As he then set about departing back to Syria, leaving Doloaspis and Petisis, two Egyptians as governors, but Petisis refused and Doloaspis became sole governor of the whole province. He then left a Greek and a Macedonian as garrison commanders in Egypt, with a garrison force of 4000 soldiers, and left a Macedonian commanding 30 galleys to defend the mouth of the Nile at Pelisium, from attacks by the sea.

With that settled, he grabbed his remaining men and left for Asia.

Sources: Anabasis of Alexander, Arrian

Histories of Alexander, Curtius

Livius.org

EDIT: Cleared up some things.