If you do arrive on the 4th, it's possible you can get tickets for the semi-finals which will be soon after but are you planning on being in Florence for the entire week? Maybe you had planned on doing something else in the meantime....? It's hard to get info but know this, you can only buy tickets here in person, not online at all. If you check back in the beginning of June, I will check and update any details on the Calcio Storico article about getting tickets, but it should be the same location since it has been for the last 5 years (not counting pandemic times when the games were suspended). The Box Office Toscana on via delle Carceri Vecchie. I'll just confirm what day they go on sale once the city posts info on that.
The day of, even if you don't get tickets, you can enjoy the parade downtown before the game: the teams deck out in their costume and the societies that dress in historical costume parade from piazza Santa Maria Novella to Piazza della Signoria and then on to Piazza Santa Croce, so you will still get to enjoy that, it is pretty neat!! The game can be watched on TV on local channels, free! So if for some reason you can't get tickets, head back to your hotel room or apartment and watch it there