I thought I’d write an update in case it may help others.
I booked my Last Supper Ticket over three months ago and I finally got to see this masterpiece this morning.
When I arrived to pick up my ticket ( 9:30AM), there was a young couple ahead of me and I heard then asking if there were any tickets available for today. The woman behind the counter checked on her computer and told them there were tickets available for 4PM.
So... If you weren’t able to book your tickets in advance, there may still be tickets available when you get to Milan.
I’d suggest getting to the ticket office when they open at 8:15AM (Tuesday -Sunday)
Edited to add...
BTW, the ticket office is located around the corner from the entrance to the Last Supper. As you face the church, look to your left where the entrance to the Last Supper is, then walk to the left around the corner, you’ll see a light grey sign on the wall which reads, “Biglietteria”.
FYI, you’re not allowed to enter with a backpack (or a water bottle); there are lockers you can use at the ticket office.
Your purse or small bag will be searched by a security guard and you’ll have to walk through a metal detector prior to entering the Last Supper “ Museum”.
Hope this help!