Quick answer: Munich Airport has a staffed luggage storage service at €5.50 per 24-hour period, with locations in the Munich Airport Center (MAC, open 5:00–23:30) and Terminal 2 public area (open 6:00–22:00).
Where to store your luggage
Munich Airport Center (MAC) — between Terminal 1 and Terminal 2, open 5:00 AM–11:30 PM
Terminal 2 public area — open 6:00 AM–10:00 PM
Cost
Storage costs €5.50 per 24-hour period per item — a straightforward daily rate rather than a per-hour charge, so it's worth it even for a few hours if you need your hands free.