You will go through immigration in Munich because Munich-Warsaw is a local (Schengen area) flight. So you must enter the Schengen area in Munich. Not a big deal especially with a US passport. I don't know if you go through securiity -- I think I did when I connected at Frankfurt from the USA but don't remember positively.
One hour is ok, though even as a somewhat experienced traveller, I would opt for a longer connection time, at least 90 minutes.