Thank you for contacting History Hub!
The National Archives at College Park holds records from Record Group 84: Records of the Foreign Service Posts of the Department of State, 1788 - ca. 1991. The series "Classified and Unclassified Central Subject Files, 1964 - 1966" may contain some description of personnel, though I am unsure if a list of the staff would be available. Their email is Email: Archives2reference@nara.gov
You may also want to contact the embassy in Cairo and see if there is any service for contacting other past personnel.
I hope this is of some assistance to you!
Hello Terry Gingerich,
In addition to Mr. Farmer’s suggestions you may also be interested in the Department of State publication called the “Foreign Service List,” certain volumes of which may be accessed through the HathiTrust Digital Library. Their catalog includes two entries with volumes dated 1965 and 1967 at https://catalog.hathitrust.org/Record/001719572 and https://catalog.hathitrust.org/Record/012408077 with staff listings for the Cairo embassy to the United Arab Republic such as the October 1965 volume. Per the Preface to that volume the lists published around the 1965-1967 time period include “chiefs of mission; Foreign Service officers; Foreign Service Reserve officers; Foreign Service Staff officers of class six and above; and consular agents. It also includes officers of the Agency for International Development, the Peace Corps, the United States Information Agency, the Foreign Agricultural Service of the United States Department of Agriculture, the Departments of Army, Navy, and Air Force, and United States Missions to International Organizations.”
We hope this is helpful. Best of luck with your research!