There isn’t a single moment that is officially designated as the beginning of the Passion. Prior to Vatican 2, there was a period named “Passiontide”, lasting two weeks and running from Passion Sunday (the Sunday before Palm Sunday) to the morning of Holy Saturday. In the West, the first of those two weeks was called Passion Week, but in the East (so I have read) the name Passion Week designates the second of those two weeks, which in the West we call Holy Week.
And then, as Fr. @edward_george1 notes, the Passion itself may be said to begin with the Last Supper, though in some Bibles the editors place the heading “The Passion” at a later point, when Jesus and the disciples leave the Upper Room to make their way to Gethsemane (Matt 26:30, Mark 14:26, Luke 22:39, John 18:1).