Turkey to most countries issues visa at the airport upon arrival. But if Egypt is one of those countries i don't know.
Better call the Turkish Embassy here in Doha.

TURKISH EMBASSY IN DOHA
Postal address:
Dafna- Al Istiqlal Street
Doha- Qatar
P.O. Box: 1977

Telephone:+974 44 95 13 00

Fax:+974 44 95 13 20

e-Mail: [email protected]

Working Hours:
Sunday - Thursday
09:00 -12:00