The rookie who pushes Tom Selleck's character off the Yankees roster was played by Frank Thomas, who went on to become one of the best MLB hitters in the 1990s and won two MVP awards.
At one point in the movie, when the pitchers refuse to pitch to Selleck's character because they don't want him to break the home run record, Selleck turns his bat the other way around and challenges pitchers to pitch to him as a sign of protest. This comes from a real-life incident a few years before the movie in the Japan Leagues when Hanshin Tigers slugger Randy Bass was on the verge of breaking the single season home run record there and pitchers refused to pitch to him. Bass similarly turned his bat upside down to protest.
At the end of the film, Tom Selleck's character is wearing the uniform of the Detroit Tigers. He's wearing the same baseball cap that he wore during his days on the television show, "Magnum P.I."