I lived in Rome in the late 60s. I never visited any contemporary military
museums there, but see this link.
http://aeroweb.lucia.it/en/museums/vigna.htm.

From visiting other museums in Italy, beware of posted times, and
double-check locally before you go. There are SO MANY OTHER THINGS TO SEE
dealing with ancient Rome that you just might pass on the modern stuff. I
recommend the Capitoline and Vatican Museums during any first trip to the
City, and a look at Hadrian's Villa in nearby Tivoli.

A possible day trip to see the beaches and war cemeteries in Anzio-Nettuno
is worthwhile, ditto the area around Cassino but don't try both in one day.

The coolest "military" stuff I saw in all of Italy was in Naples, at the
"National Museum" downtown. The collection of gladiator helmets and armor
there will blow you away: it all comes from the excavations at Pompeii.
That would be a looooooong day trip, about three hours one way by train from
Rome, as I recall.

In Rome itself you should walk from the Colliseum to Piazza Venetzia, where
Mussolini spoke from the balcony, then up the Capotline Hill steps, and go
around to the right and "back" for a magnificent view of the main Forum.
Don't miss walking across the Island In the Tiber and checking out the
remains of the Circus Maximus. Empty but still awesome in size.

I could go on and on but hope this helps.

Brian