Flying in those times was a luxury, from what I understand. Would black passengers be segregated to specific parts of the plane? Did this change depending on airline or route?
Airline travel was Federally-regulated and the Civil Aeronautics Act of 1938 prohibited discrimination of airline passengers by race. Therefore, the airplane cabins were integrated. There was no "white section" and "black section" inside the airplane cabin.
However, municipal airports were governed locally and there were airports in the South that had segregated dining areas, ticketing counters, etc. Municipal airports were eventually integrated by the Federal government withholding grant funds used for building or expanding airports if the airport was (or was planning to be) segregated.
For more details, see Jim Crow Terminals: The Desegregation of American Airports by Anke Ortlepp.