In "To Kill a Mockingbird" does Jem's behavior toward Scout at school seem realistic?
It think that it is pretty natural to be embarrassed by a little tag-along sister, especially if you are a pre-teen guy. Guys tend to give each other a hard time over everything, and Jem was probably worried that Scout would be an annoyance, or reveal something embarrassing about their home life, or the games that they played. Scout's lack of tact in general makes this a valid concern. He tries to take precautions. He tells her that she "was not to bother him..not to approach him with requests to enact a chapter of Tarzan and the Ant Men, to embarrass him with references to his private life...in short...to leave him alone." This seems like pretty stereotypical bigger brother stuff; you have to remember that Scout and Jem had quite a few years in-between them, so being seen at school together wouldn't be quite as okay as if they were closer in age.
Jem isn't completely rude; he does condescend to interact with Scout when necessary. For example, when she is beating poor Walter Cunningham up, Jem intervenes. And at the end of the book he helps to get her to and from the pageant at the school too, so he is involved, just not like they are at home.
If anyone has ever been embarrassed by a sibling or parent, it is easy to find Jem's behavior completely normal.