Upvote:1
United and Lufthansa are both Star Alliance, so checking the bags through should be no problem, simply ask for it when checking in at O'Hare.
Cyprus is not in Schengen, so you will not have to enter the Schengen area in Munich. You can stay airside in Munich und transfer to the Lufthansa flight directly. No need for customs or immigration.
Immigration and customs will happen in Larnaca.