Well, the simple answer is that the Achaemenids did construct new buildings, and plenty of them, but neither they nor anything else from the Egyptian Late Period is quite as flashy or tied to famous historical events the way that buildings from Egypt's richer and more powerful phases is. So Late Period sites just don't get as much attention. Due to their historical proximity to the Ptolemies and Rome, Late Period projects were also just more likely to be built over by those later conquerors, further diminishing their visibility.

In the case of the Achaemenid Persians, there is another complication. You're not going to find something built in the style of Persepolis or Pasargadae in places like Egypt, Syria, Lydia, or Babylon. In conquered territory where there was already an established administrative structure, the Achaemenids were content to just let the existing system perpetuate and pay taxes. Even under Persian rule and using Persian funds, the Egyptians kept building temples and crafting art in traditional Egyptian style. So, nothing stands out as overtly Persian until you start reading the cartouches that identify the Pharaoh as Cambyses, Darius, Xerxes, or Artaxerxes. Egypt was also incredibly dense and ancient already by the Achaemenid conquest, meaning that Persian patronage was often directed toward restoration rather than new buildings.

On top of that, the Egyptians were often hostile to Persian rule. Darius II was the only Achaemenid not to face and Egyptian revolt, and only because he happened to die just as the rebellion that briefly restored Egyptian independence started. In some cases, this resulted in Persian-sponsored building projects to endear themselves to the Egyptians. In others, it led to harsh punitive measures that stripped the Egyptian elites of their independence and destroyed existing buildings, which leaves us with a mixed bag of Persian period evidence. The gradual replacement of Egyptian elites with Persian overseers also means that we cannot rely as heavily on traditional Egyptian burial customs to provide evidence. Persian officials were not mummified and entombed in Egypt, limiting the high status graves for the period to lower Egyptian officials.

There are plenty of inscriptions, stele, and papyri from the period, but I'll focus on buildings per the OP. Archaeology of Empire in Achaemenid Egypt by Henry P. Colburn is a key resource for this topic, and grew out of his PhD dissertation, which can be freely accessed as a pdf from the University of Michigan. The traditional Egyptian capital at Memphis was adopted as the seat of the Persian Satraps, and due to its antiquity the city largely showcases Achaemenid maintenance of existing palaces and temples. However, the city's foreign quarters, largely built around the Temple of Ptah, were constructed to accommodate the influx of non-Egyptians under Persian rule. Herodotus references a Tyrian Camp in the city by the 450s BCE, and the Carian and Greek quarters first appeared under Persian auspices.

The Persians also expanded several existing structures in Memphis. A military garrison, possibly Herodotus' Tyrian Camp, was installed in the so-called Palace of Apries, and contrary to the Greek tale of Cambyses slaughtering the sacred bull, the Temple of Apis was greatly expanded (including a Stele commemorating Cambyses honoring Apis). A funeral for the contemporary Apis bull was recorded on Stele in 413 BCE, and a hoard of Athenian coins from the time was buried in the foundations of one of the expansion projects, indicating that the project was probably commissioned by Darius II as part of those festivities. It would have continued into the early reign of Artaxerxes II but likely only finished under one of the newly independent Pharaohs after 401, who carved over Persian cartouches in ongoing building projects.

If you want grander Achaemenid projects, its best to look west, away from the Nile, to the oases in the Egyptian desert. Not only were these sites more neglected during the preceding periods of Libyan-Egyptian and Nubian rule, but they were some of the most hotly contested regions in the Achaemenid period. The strongest evidence for support of rebel Pharaohs often comes from the oases as well, which prompted a concerted Persian effort to build up good will and official presence to deter rebellion.

One of the earliest examples of this was the Dakhla Oasis, possibly the seat of the rebel Pharaoh Petubastet III at the outset of Darius the Great's reign. Petubastet began the construction of a new Temple of Thoth, but Darius oversaw its completion and the construction of several other temples at Dakhla. The town around the oasis was also heavily expanded in the early-5th Century BCE, and pottery styles at the site indicate that this expansion was for Iranian settlers rather than local Egyptians or Libyans, a sign of increasing Persian presence in the oases.

Though not a building per-se, Darius the Great also ordered the construction of one of the most impressive infrastructure projects of the age: The Grand Canal. A sort of proto-type the Suez Canal today, this was a actually a series of canals linking the Nile to the salt lakes in the Sinai Peninsula and linking the lakes together before ultimately connecting to the Red Sea in the Gulf of Suez. To commemorate its completion, Darius erected a unique quadralingual monument near modern Suez using Egyptian hieroglyphs alongside the traditional cuneiform languages of the Achaemenid inscriptions (Old Persian, Akkadian, and Elamite). Thanks to the shallow draft of most ships at the time, and the depth of the Nile, this created a shipping route between the Mediterranean and the Indian Ocean for the first time. Unfortunately, it was also prone to silting shut at the Red Sea end and expensive to maintain. Later Achaemenids or independent Pharaohs gave up on the full connection at some point, only to be re-opened by Ptolemy II.

The Kharga Oasis provides the most impressive examples of Achaemenid building Egypt home to the Hibis Temple of Amun. The modern ruins include many Ptolemaic additions, but the core was initially constructed by the Persians. Conventionally, the Hibis Temple is identified with Darius the Great, partly because he was such a prolific builder that otherwise unlabeled "Dariuses" are attributed to him, but there is a chance that the construction in the Kharga Oasis was under Darius II. Several of the cartouches from the Persian period were left empty, probably reflecting a break in construction during an Egyptian revolt. Most of the later work on the site came long after Darius the Great, and one of the cartouches features a unique Horus Name, Menekhib. The Egyptian inscriptions firmly tied to the original Darius use the Horus Name Wernebmeryshemau. Sometimes this is interpreted as Darius the Great unconventionally using two Horus Names because neither Xerxes nor Artaxerxes I seem to have adopted full Egyptian titles, but its always possible that Menekhib refers to Darius II.

Many additional structures near Kharga were also constructed under the Achaemenids, but the most notable are three qanats, a sort of underground aqueduct first developed in Iran.

Finally, there are written records of Persian building projects in Egypt that do not show obvious archaeological remains. Most notably, the Temple of Yahweh on the island of Elephantine at Egypt's southern border. Following the Babylonian conquest of Judah, many Jewish refugees fled to Egypt, and a large number of them were resettled on Elephantine as military garrison. They built a new temple on the island to continue on with their tradition rites despite the destruction of Jerusalem.

In the late 5th Century, tensions over property lines, religious sacrifices, and royal patronage with the much older Temple of Khnum on the island escalated to open conflict between the Jewish garrison and the Egyptian priests. The Jews historically had the support of the Satrap, but communication was slow and the Persian garrison nearby backed the Egyptians and razed the Temple of Yahweh. In the aftermath, Satrap Arsames sided with the Jews, and made provisions to rebuild their temple with imperial funds. The project barely got off the ground because Egypt seceded just a few years later, and the new pharaohs cut off funds.