What is the structure of the story "The Night the Ghost Got In"?
I think that James Thurber would be amused to hear that you are looking for the introduction, rising action, climax, falling action, and denouement in his stories. He once wrote that his stories are about "the confused flow of his relationships with six or eight persons and two or three buildings"--and "confused" is the key word.
Nonetheless, "The Night the Ghost Got In" does contain most of the standard elements of plot.
Introduction: The story begins at 1:15 A.M., with the sound of a "rhythmic, quick-cadenced walking around the dining-room table."
Rising Action: The narrator awakes his brother, Herman, and his mother. His mother wants to telephone the police, but the only telephone is downstairs, together with the ghost. She flings a shoe through a neighbor's window; the neighbor wakes up and calls the police.
Climax: One could argue about identifying the climax of the story, but I would say that it is when the police arrive and they hear some noise from the attic. That's where Grandfather sleeps, and he is suffering from delusions. He imagines that the police are Civil War soldiers, and he shoots an officer in the shoulder.
Falling Action: Although the wounded policeman would like to investigate the attic, his colleagues convince him to leave the matter to rest. The officers leave the house.
Denouement: At breakfast the next morning, Grandfather "was fresh as a daisy and full of jokes." He asks the family, "What was the idee of all them cops tarryhootin' round the house last night?" Thurber concludes, characteristically, with a note of doubt and confusion: "He had us there."