Yes. He starred in small parts during the late 30s, but by the end of the decade he had a large role in Knute Rockne: All American (where he got his 'the Gipper' nickname) and was very much in the public eye as a young leading man. What was probably his most famous role came soon after in 1942 in a very toned-down adaptation of the racy novel King's Row. That marked the peak period of his career, which never quite recovered after his war service. At the highest point in his acting career, think maybe Joseph Gordon Levitt-ish level of fame.
Ronald Reagan was in the radio, movie, and television business from 1937 to 1957, in which he appeared in 52 movies. He was probably most noted for Knute Rockne—All American (1940), a movie of a Notre Dame's Football coach. So I would say that Reagan was famous before being president and was a successful actor.
Soure: http://millercenter.org/president/reagan/essays/biography/2
He got his nickname "the Gipper" from his role in the film "Knute Rockne, All American", which he later revived during his presidential campaign. (As in "Win one for the Gipper!) So people at least knew of his acting career during his political career, but you want to know if he was popular before.
I seem to remember him finishing fifth in a stardom contest of some sort just prior to his military service, I'll find a source for that in a couple hours after I get out of class. Edit:in the top right corner you can see Reagan's name. This paper was published in 1941, well before any known Reagan political activity. I'm also fairly sure the newspaper is Australian, which speaks as to some sort of intercontinental popularity. /edit
He was president of the Screen Actors Guild, for what that's worth. I don't think that particularly speaks to his acting skills, but I would say it lends credence to the idea that he was indeed widely known before his political career as an actor.
According to wikipedia at least, his army service included stints in a film unit filming propoganda, so before the end of WW2 he was known enough as an actor that the army wanted him to film stuff and not fight. Hopefully someone will provide a better source than wikipedia for me?