SABA'A AL THABIT, Egypt—Self-styled strongman Al Sayed al Essawy had an idea for lifting his country out of its post-revolution economic funk: Fight a lion.

Which is why on Saturday Mr. Al Essawy stepped into a steel cage with a 660-pound lion here in the middle of a wheat field in this farming hamlet. He glared at the lion and bared his teeth. He carried a "shield" made out of an old satellite dish.

Addressing...