Combine first 4 ingredients in a small saucepan. Bring to a boil, and cook 5 minutes. Remove from heat; cover and let stand 30 minutes.
Coat a medium nonstick skillet with cooking spray; place over medium-high heat until hot. Add onion and vinegar; cook 20 minutes or until deep golden, stirring frequently. Remove from heat; stir in fig mixture and cheese. Set aside.
Place 1 phyllo sheet on work surface (cover remaining dough to keep from drying); lightly coat with cooking spray. Working with 1 phyllo sheet at a time, coat remaining 7 phyllo sheets with cooking spray, placing one on top of the other. Place a sheet of plastic wrap over phyllo, pressing gently to seal sheets together; discard plastic wrap.
Spoon onion mixture along 1 long edge of phyllo, leaving a 2-inch border. Fold over the short edges of phyllo to cover 2 inches of onion mixture on each end.
Starting at long edge with 2-inch border, roll up jelly-roll fashion. (Do not roll tightly, or strudel may split.) Place strudel, seam side down, on a jelly-roll pan coated with cooking spray. Score diagonal slits into top of strudel using a sharp knife. Lightly spray strudel with cooking spray.
Bake at 350° for 30 minutes or until golden brown. Sprinkle powdered sugar over top. Serve warm.