In a way, yes. But not as much as they had famous sports stars and actors.

Music was a big part of Roman life and was part of the pantomimus which encompassed dancing, song and instrumental music. It was customary at funerals, games, sacrifices and many social occasions. The thing the Romans did not do well when it came to music was originality. Many of their musical attributes were taken directly from Greece. They did hold competitions though, all over Roman lands and Greece, where many famous musicians competed, even emperor Nero at one point. These competition winners could be a kind of "rock star".

There were certainly famous actors. Like Quintus Roscius, who was so well renowned that even students under his tutelage were pretty much guaranteed success and fame, despite many expressing distaste for actors.

Then of course there are the Gladiators and Charioteers, many of whom became insanely rich and famous.