There is actually more than one answer to this question, depending on the city you are talking about.

It is entirely true that the names you are reading are Greek sounding and this is because our knowledge of Ancient Egypt, especially that which existed before we were able to decipher hieroglyphics (enabled greatly due to the work on the Rosetta Stone in the early 19th Century), came either from Greeks or from Hellenized Egyptians.

Egypt was conquered and Hellenized in the time of Alexander the Great and his successors the Ptolemaic dynasty, after which the period of purely Egyptian culture ended and scholarly work became mixed to a large degree with that of the Greeks.

While certainly ancient Egyptian language and hieroglyphics were still known at the time, the Greeks and Hellenized Egyptians, and significantly the later Romans, would have read the Greek texts and considered the Greek names to be current for their time. These texts would have made their way down the ages to scholars and would have been accessible to them due to wide knowledge of Greek, especially in the Byzantine Empire, the Arabic empires, and later in Renaissance Europe.

One of the major contributors to the knowledge of Egypt that has come down to us that did not require knowledge of hieroglyphics was from the Hellenized Egyptian priest Manetho who wrote the Aegyptiaca (History of Egypt) in Greek.

Firstly, the place names of pre-Ptolemaic Egypt are sometimes Hellenizations of the existing Egyptian names. For instance, the city of Memphis, which was capital of Egypt during the Old Kingdom period had a number of names, but by the point of the New Kingdom it was known as mn-nfr (Mene-fer) "enduring and beautiful" which was Hellenized into Memphis.

Secondly, the Greeks did not stop at making Egyptian names more "Greek-sounding", they actually completely changed the names to make them actually Greek in meaning. The city known best as Heliopolis was known as I͗wnw (Iunu) to the ancient Egyptians and meant "the pillars". The Greek name comes from the fact that the city was the center of worship of Ra and the Greeks associated this worship with the city. Since Ra was associated with the Greek sun god, Helios, the name of the city became "City of the Sun" or Heliopolis in Ptolemaic Egypt.

The same goes for Herakleopolis which was originally known as nn nswt "Child of the King". Similarly to Heliopolis, Herakleopolis was the seat of a cult of the Egyptian god (Heryshaf) who the Greek rulers of Egypt associated with Hercules.

This practice of associating gods with Greek gods is known by the later Latin term interpretatio graeca and was used by writers like Herodotus and those that followed to try to relate other culture to Greek culture as they understood it, and the Romans would do this in reverse when they made links between gods like Jupiter and Zeus, which in turn permitted syncretizing religions.

Greece, but particularly Rome later on very much favored this process and in the later Roman empire there would be many cults of Egyptian gods who had been Hellenized in the East by the Greeks first, and then adopted to some degree by the Romans who took over those areas later on. Isis and Serapis would be examples of either an Egyptian god popular with the Romans and Greeks or in the latter case, a completely "invented" god combining elements of a number of existing gods that was installed by the Ptolemaic dynasty to give itself some legitimacy.

Lastly, of course, there were some cities that were founded by the Greeks themselves directly, notably Alexandria itself, and they would have started out with Greek names and were more or less Greek cities with Greek populations.

Now, you did ask about how the Arabs might refer to the same cities, and of course, they do have their own names for these places.

However, in many cases, the Arabic names refer to the places as they might have been found at the time of the conquest of Egypt. In those time periods, many of the old cities were still known, but only as ruins having been eventually abandoned in favor of Alexandria in the later Roman and Byzantine periods. Their demise would also have been affected by the Christianization of Egypt and the eventual reduction and end of the cults of the gods who had been the centerpieces of those cities.

Heliopolis itself is known as Ayn Shams in Arabic, which means "Eye of the Sun" a clear reference to what Heliopolis was known for, but it was a ruin at the time of the Arab conquest and its materials were used to help build Cairo.

Memphis shared something of the same fate, and it's ruins were also used to build Cairo as well as raided for other sites. It is simply known as Manf.

You will definitely want to read more about Manetho and Herodotus as well as the Ptolemies in how Hellenization came to pass in regard to Egypt. It was part Greek scholarly interpretation, and part cultural assimilation which resulted in the names we know of today that sound Hellenized. The Ancient Egyptians before the Greek conquest would not have used those names for their own cities unless they had some commerce with the Greeks before the conquest.