How did Ptolemy's empire function?

Ptolemy synthesized Greek knowledge of the known Universe. His work enabled astronomers to make accurate predictions of planetary positions and solar and lunar eclipses, promoting acceptance of his view of the cosmos in the Byzantine and Islamic worlds and throughout Europe for over 1400 years. I don't know if that directly answers your question though.

They lived in solitude, and under Alexander the Great's rule

Other Information if Needed

They never really became Egyptian, despite ruling over Egypt,

before falling to the Roman