The rising action is the events that build to the point of greatest interest. The narrator watches the man sleep for 7 nights, leading up to the 8th day where things change.
The climax is the turning point of a passage in which questions are answered. On the 8th day of watching him, the narrator gets too fed up with the eye and ends the old mans life. "I dragged him to the floor and pulled the heavy bed over him."
The falling action occurs after the climax and starts to wrap up the story. The police come to the house to see what is going on. They have heard complaints of a scream.