Exposition: In "A Rose for Emily," Faulkner's approach is not linear, as he moves back and forth in time. Although the story begins with Miss Emily's funeral, the exposition occurs shortly thereafter when we meet Miss Emily through the narrator’s eyes and are told something about her background, her family history, and her eccentricities. Two paragraphs after the opening, Faulkner begins the exposition with,

Alive, Miss Emily had been a tradition, a duty, and a care.

Climax: Despite its position at the very beginning of the story, the climax occurs when Miss Emily dies. The story opens with the townspeople attending Miss Emily's funeral.

Rising action: This occurs when Miss Emily goes to the pharmacist to purchase poison and we learn about the stench emanating from her home and that Homer has left her.

Falling action: Immediately after Miss Emily’s funeral (the climax), the townspeople are able to get into her house for the first time in years. Their discovery there wraps everything up, leading to the resolution.

Resolution: Once inside Miss Emily’s home, the townspeople discover Homer's skeleton and a single long grey hair on the pillow beside his head. This resolves everything that we have learned previously in the story, as we realize that the poison was not for rats, but for Homer. We conclude that Miss Emily killed Homer, probably because he threatened to leave or because she was just afraid that he would leave her. Once he was dead, she lived with his decaying corpse as if they were man and wife. There is symmetry here to her earlier refusal to allow the locals to take the body of her deceased father for burial when she wanted to remain with her father even after his death.

We do not necessarily have to consider the story in chronological order in order to assess these elements.  In fact, the story is not presented to us in chronological order, so I would argue that we ought to honor the order of events in which we do get the story when analyzing it.  As a result, descriptions of Miss Emily's funeral, the town's feelings about her during her life, and her taxes are all exposition.  

The rising action begins with the description of the smell that once emanated from Miss Emily's home, thirty years prior to the tax conflict and just two years or so after her father's death.  Next, we learn about her father's belief when Emily was young that no one was good enough for her.  Then he dies, leaving her all alone, something that she is clearly uncomfortable with because she hoards his body for days before allowing people to take it away and bury it.  This is also an important instance of rising action because Emily's odd treatment of the dead is a clue to understanding the story's climax.  Emily gets sick, recovers, meets Homer Barron, buys arsenic, refuses to tell the druggist what she needs it for, and buys wedding gifts for Homer, before he disappears into her home for good, never to be seen again: This all falls under the characterization of rising action, as is the description of her "iron-gray" hair.  

Finally, Emily dies, and after her funeral, townspeople know that the door to one room upstairs "would have to be forced open." It had not been opened for some forty years.  In the story's climax, there is the "violence of breaking down the door" and the discovery of Homer Barron's decayed body, surrounded by his bridal suit and the gifts Miss Emily purchased for him.  This is the moment of the most tension in the story: we learn that she has hoarded Homer's body just as she did her father's.  

In the story's falling action, the narrator describes Homer's body, its attitude, and final posture, as well as the fact that it has essentially rotted so much that it "had become inextricable from the bed in which he lay."  In the story's resolution, we learn that a long piece of Emily's easily identifiable (as a result of the earlier description in the rising action) hair is found on the pillow beside Homer's.  At this point, what we might have suspected is confirmed: Emily murdered Homer in order to prevent him from ever leaving her, as he apparently briefly did when her awful family came to visit at the request of the townsfolk.  

This story is told almost backwards with its use of flashbacks. So the way to examine it's plot sections is also backwards.  The exposition of the story would be when the author introduces her father and we see his personality and her background.  We know the characters involved and the conflict.  Emily is too good for any man, according to her father, so he keeps her from dating/marrying.  Then he dies, which is another conflict for her--being alone.  This carries on throughout the story.  She does not want to be left.

The rising action involves most of the rest of the storyline including the town's attitude towards her and her fling with Homer.  Even the part where she buys the arsenic and the house smells something awful.  The town even spreads lime around the house to help keep the smell down.

The climax is not until the last few lines of the story when we find Homer's body and one of her gray hairs on the pillow next to his corpse.  We realize that she had poisoned him so he wouldn't leave, (and that was the awful smell earlier) and that she has been lying with him ever since.

The falling action is about a decade before she dies when they try to get her tax money from her.  She holds them off, though.  And the resolution then is the really at the beginning when she is introduced at her own funeral.