Hi Roni,
Since your travel dates are coming up quickly, it may be best to have alternate plans in place. Free is always good, but of course you'll have lots of competition. And, since your time in Rome is a shorter stay, you probably want to maximize your travel.
You mentioned consideration of a tour. So that may be a good alternative.
Here is the link for some helpful info about The Papal Mass and other tidbits http://www.papalaudience.org/papal-mass
Official site of the Vatican for tickets and other important information http://www.museivaticani.va/content/museivaticani/en.html
The Vatican does have its own guided tours. If you want to attend Mass celebration, you have to do it on your own. For the most, up-to-date information, check their official website linked above.
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni.html#lnav_info
There are also very good, guided tour groups:
https://theromanguy.com/tours/Vatican (between 14 to 18 peeps) on avg.
https://www.walksofitaly.com/vatican-tours/ (up to 20 peeps)
https://darkrome.com/vatican-tours (up to 20 peeps)
It appears that (as of this writing), all three tour companies have availability for a Saturday morning or afternoon booking of the Vatican.
You won't make a tour Friday afternoon with TheRomanGuy - last time for a regular tour is 2:30 PM; WalksofItaly has 7 AM tour; DarkRome starts at 8 AM
Of course, it's always best to check each website for their most up-to-date information on schedules and policies. Ask about discounts for early booking or combined tours discounts. They may offer a discount if you book more than one tour with a company.
There are policies for both the Vatican, The Sistine Chapel and St. Peter's Basilica.
Dress code - strictly enforced; especially as we come into the warmer months. Modest clothing in Churches. No shoulders or knees bared for both women and men. I don't know about policy for children, but to be on the safe side.
Security and bag check. No big bags or luggage allowed. Small to medium sized bags were ok'd by guards, but a busier time and huge crowds may change allowing backpacks. Best to have a cross body or messenger style bag. Water bottles must be put away in bags.
There is a 'coat room check' - free - but that will definitely impede your going to St. Peter's after The Sistine visit. You would have to retrieve your stuff and wait in line again for St. Peter's. (It is free to enter at any time even if you don't go to The Vatican.)
No cams, phones or talking in The Sistine.
Rome is great! Have fun.