For full disclosure, I worked on a project for this museum, so I am biased.
This museum is small, but book a tour with the lovely guide Chiara and she'll take you beneath the Duomo. Most of the artifacts from this site have gone to the Archaeological Museum of Umbria, but they couldn't move the real treasures.
You can see three layers of human occupation. The Etruscans, the Romans, and the Medieval period.
You can see the area where 3 popes were elected. You can walk on a Roman road that still has wheel marks in it. You can see massive Etruscan walls.
My personal favorite part is the roman house where there's a burn level that shows the destruction of Perugia by Augustus.
For a museum that just has Chiara running it, it is impressive. That's why you must book a tour and for your convenience, Chiara has spent the last few years learning English (she says her English is bad, but she's just modest).
It can be cold underground, even in the summer, so maybe bring a cardigan too.
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