I have a question regarding flying into Munich from the US, and catching another flight to Italy 2 hours later.

I will be flying into Munich from the US on UA landing about 9:50 am. I am thinking of catching an 11:50 am flight to Rome using Alitalia. I am pretty sure the UA flight lands in terminal 2 and the Alitalia flight departs in terminal 1. Will I have time in Munich to go through immigration, collect my bags, check in myself and my bag for my flight to Rome, and get to the Alitalia gate on time?

The two flights are booked seperately, so there will not be an automatic transfer of baggage.