The movie was "Arashi wo Yobu Otoko" (Man Who Causes A Storm) which was released in December 1957. It's about a young ex-con named Shoichi (played by Ishihara) who finds a job as a drummer in a Ginza nightclub and proceeds to claw his way up the ladder. The big scene is when Shoichi sings the main tune while he drums up a storm despite having injured his hand.
Written by Umetsugu Inoue（井上梅次） and composed by Seitarou Omori（大森盛太郎）, the song sold 630,000 copies after its release in February 1958. There wouldn't be an Oricon ranking system for another decade, but if there had been, this song probably would've been a No. 1 easily.
The above is the trailer for the movie. Seeing Ishihara's two different types of cool in the 50s and in the 70s/80s rather reminds me of Clint Eastwood. Perhaps his brother, former Tokyo governor Shintaro Ishihara, could've felt the same way when he met Eastwood himself a few years ago when the director came to Japan.
And in one of the recent Kohaku Utagassen specials, Ishihara's old kohai and buddy from the Ishihara Gundan, actor/singer Hiroshi Tachi（舘ひろし）, gives his own tribute to The Big Man.
(The performance starts at around 5:20 below.)