When did Arrowsmith get popular?

If you mean the book Arrowsmith - it was published in 1925 and recieved the Pulitzer Prize in 1926 - by which time you could consider it popular.

If you (more likely) mean the band Aerosmith:

The band was locally popular almost as soon as they formed in 1971, playing shows in the Boston area. They hit the big time when they signed with Columbia records in 1972 and started releasing a string of extremely successful records and albums staring with their debut album Aerosmith, followed by Get Your Wings (which went platinum). They achieve international stardom with the release of their 1975 album Toys in the Attic.

As an interesting note - when band member Joey Kramer first suggested "Aerosmith" as a name for the band, many of his bandmates thought he was referring tothe novel "Arrowsmith" they were required to read in high school English class. "No, not Arrowsmith," Kramer explained. "A-E-R-O...Aerosmith."