When does 24 hour antigen test clock end?

The correct answer is when you cross the border, but who knows at what point exactly they will check your test status. Maybe they'll check it onboard the train upon crossing the border, maybe they'll check it when you get off the train in Bologna. Or maybe, but very doubtful, there will be enough cooperation that the German Bahn staff will check it before you board in Munich, as the airlines do.

A friend of mine was recently refused boarding on a plane because her test would have expired one hour before the plane was scheduled to land in Italy. So I'd give yourself some wiggle room and maybe not take the test earlier than 5:00 pm the day prior to your 4:00 pm arrival just to be safe. Don't make it too tight and give them a reason to complain, is my opinion.

To put it simply, it ends 24 hours after you take the test, like a timer, independent of any time zone changes.

You have a bit of a wrinkle, from the standpoint of the airline, you are going to Germany, not Italy, so they will check your documentation for Germany and apply any German covid requirements. So to board your flight, be concerned about those rules.

Normally, entering Italy by air, only the airline will check your test result, as they have been delegated that responsibility, they do not bother to check in Italy. Arriving by train? who knows, I doubt they set up internal border controls, you are still required to meet the requirement, even if no one checks. To be 100% safe, you might be able to get a quick test in Munich, that would answer all concerns.

Of course the OP might be entering Italy by train from Munich days or weeks after arriving in Europe on a flight.

We arrive in Munich from the US via United on March 16, but depart from Munich via train to Italy on March 19. As other have noted the safest bet is to get the test as late as possible. I am so glad that I found this bit of information as otherwise I likely would have simply scheduled the test for Friday morning perhaps.

Use this tool
Current rules are valid thru 1/31

https://infocovid.viaggiaresicuri.it/index_en.html