We also love the Sistine Chapel and this past August it was on our boys' list of "want to sees." Instead of going first thing, we went to the Vatican Museum at the end of the day, entering about 15 minutes before the ticket office closed. We then had plenty of time to wander through to the Chapel, sit and listen to the Rick Steves tour (podcast) on our devices and visit the gift shop. It was refreshing how empty it was, no problem getting a seat and no rushing or jostling. Highly recommend going late in the day!

If you are worried about getting a "stiff" neck, bring a mirror & hold it by your waist
and you will be able to see without actually looking up. Good luck !