The German travel system is pretty good and not too expensive. Just stay somewhere interesting that’s not too far away, but doesn’t have a big stadium with games scheduled.
Get in early on the train, enjoy the beer, the city and the game, then either back in time for the last train or do an overnighter in a pub or club and get a train back the next morning.
They even have a ticket that costs €49 for a calendar month for the whole of Germany, but only valid for the local transport like buses, U-Bahn, S-Bahn and regional trains, so you can’t get on the fast trains like ICE/IC/EC, but the regional trains aren’t bad. Takes like 1 hour 10 minutes Hamburg to Bremen, 1 hour 20 Bremen to Hanover, 55 minutes Karlsruhe to Stuttgart, 45 minutes Augsburg to Munich etc. Sometimes doesn’t work out so well like if it takes like 4+ hours on the regional trains from Hamburg to Berlin compared to 1 hour 45 minutes on the high speed, but if you stay somewhere an hour or so away from the venue with an S-Bahn or regional train connection, that ticket does you for all these journeys, as well as all the local transport backwards and forwards to the airport, and inside the cities.