Fiorello!, a show about New York City mayor Fiorello LaGuardia, is fabulous. Check it out, even if you are not interested in musical theater.
Two of the best tunes are Little Tin Box and Politics and Poker. The show straddled the 1950s and 60s: According to the notes on YouTube–which are from Wikipedia–it opened on Broadway in November 1959 and ran until 1961. The original cast included Howard DaSilva and Tom Bosley, who won his greatest fame as Mr. Cunningham on Happy Days.