Combine first 3 ingredients in a large zip-top plastic bag; add steak to bag. Marinate 3 minutes, turning once. Remove steak from bag; discard marinade. Sprinkle steak with 1/4 teaspoon salt. Heat a large skillet over medium-high heat. Add butter and oil. Add steak; cook 5 minutes on each side or until desired doneness. Let stand 2 minutes. Cut steak across grain into thin slices.
Return pan to heat. Add juice, garlic, and spinach; sauté 1 minute. Remove from heat; add remaining salt.
Spoon 2 tablespoons spinach mixture into each pita half. Place 1 tablespoon onion, 3 cucumber slices, and 1 1/2 ounces steak in each pita half; sprinkle 1 tablespoon cheese in each pita half.