Approximately 300 BC to 30 BC, or up to 642 AD.
In the context of the sciences, the Alexandrian period refers to a phase associated with the Musaeum at Alexandria - a institution which included the famous Great Library. Founded supposedly circa 300 BC by Ptolemy I Soter and patronised by his dynasty until the Roman conquest, the Musaeum was a major centre for Hellenistic learning where the arts, philosophy and natural sciences flourished. These intellectual developments are collectively known as the Alexandrian school.
The period is commonly considered to have ended with the Roman annexation of Egypt in 30 BC. However the Mauseum continued to operate for a while (though gradually replaced by the Serapeum), and the city remained a centre of Graeco-Roman learning for centuries. Consequently, at an extreme, some writers puts the end of the Alexandrian period as late as the Muslim conquest.
The Marxist.org passage you cited comes from the works of Engels; I'm not sure why the dates he gave are off by a millennia, but perhaps he meant to say 7th century.
Further Reading: https://hekint.org/2017/01/22/philosophy-of-science-and-medicine-series-iv-alexandrian-period