Combine mustard and cider in a small bowl; stir well.
Heat a large nonstick skillet over medium heat. Spread each tortilla with about 1 1/2 teaspoons mustard mixture.
Place 1 tortilla, mustard side up, in pan. Arrange one-third of cheese slices over half of tortilla; cook 1 minute or until cheese begins to melt.
Arrange one-third of apple slices over cheese; top with 1 cup arugula. Sprinkle with 1/4 teaspoon pepper. Fold tortilla in half; press gently with a spatula.
Cook 2 minutes on each side or until golden brown. Remove from pan.
Repeat procedure twice with remaining 2 tortillas, cheese, apple slices, 2 cups arugula, and 1/2 teaspoon pepper. Cut each quesadilla into 4 wedges.