Since you're starting from within Schengen, you'll do passport control - specifically exit control - in T3. Someone else will need to confirm this, but I believe that you can stay airside (meaning not have to re-clear security) connecting from T1 to T3. I have very little experience connecting at FCO, though.

Are you flying on one ticket, or did you book separate tickets for each leg of the flight? If you're on one single ticket with multiple segments, you should be able to check your bags in Germany and not worry about them until you get to Japan. If that's the case, I'd think 2 hours should be enough.

Are you flying on a European or US passport, or another entirely? The only thing I can think of that would jam you up is if you have issues with your passport/visa for Japan.