There is really only one character in chapter 24 of To Kill a Mockingbird by Harper Lee who explicitly does any running, and it is not Atticus Finch.
Most of the chapter is dedicated to the Ladies' Missionary Society meeting that Aunt Alexandra is hosting in the Finch home. Of course the entire meeting is an ironic situation, as these quite prejudiced women are happy to give their attention, money and sympathy to a tribe of black people halfway across the globe, but they treat the black women in their homes and the other black people in town as second-class citizens.
This meeting is also noteworthy because of poor Scout's attempts to be a lady in order to please her Aunt Alexandra and even Miss Maudie. She actually does pretty well, though she does better at the end of the novel after Atticus appears unexpectedly at the house.
It is true that Atticus arrives out of breath; however, there is nothing that says he has been running. If this is what your question refers to, the answer is that Atticus was out of breath because he has just heard the news that Tom Robinson is dead and has hurried home to get Calpurnia and the car so they could go see Helen Robinson and tell her the news before she heard it from some other source.
It is more likely that your question refers to Tom Robinson, who, Atticus announces,
just broke into a blind raving charge at the fence and started climbing over. Right in front of [the guards]—”
“Didn’t they try to stop him? Didn’t they give him any warning?” Aunt Alexandra’s voice shook.
“Oh yes, the guards called to him to stop. They fired a few shots in the air, then to kill. They got him just as he went over the fence. They said if he’d had two good arms he’d have made it, he was moving that fast. Seventeen bullet holes in him.
Tom Robinson was found guilty for something he did not do and, despite Atticus's assurance that they had a chance to have this sentence overturned on appeal, he had no hope that he would ever be a free man again. It is this fear and feeling of certainty which prompts Tom Robinson to make a desperate run for the fence of the prison yard. He was running for his life, and he knew he would either make it and be free or be killed and thus put out of his misery. He did not make it.