The narration that Faulkner adopts is actually a key technique for lots of Gothic writers - the way that time is treated and conveyed to the reader does a lot to create suspense and irregularity as well as heighten the sense of eventual horror when it is revealed. You might want to consider how such novels as Wuthering Heights and Frankenstein adopt a similar irregular approach to narration and time, using distancing devices such as frame stories and flashbacks and multiple narrators to convey the uncertainty and heighten the terror of the story.
However, in this short story, Faulkner starts with the end in a sense - the death of Miss Emily - and then uses a series of flashbacks to paint the background to the grisly end of the story - to highlight another key theme of the tale, which is the disintegration of the values and social order of the Old South and its replacement by a more modern set of values that does not recognise the importance of the social order in this new world. Starting off with the end of the story and then going back to the beginning creates a sharp contrast between how values were and how they are today, emphasising the massive change that occurred and highlighting the way in which Miss Emily in a sense ended up a stranger and an isolated figure in her own world, as she still clung on to the old-fashioned values and approaches of the Old South that had become anachronistic.
Don't forget that "A Rose for Emily" is considered both a Gothic short story and a murder mystery. The events in the narrative are not told chronologically, but rather, by a series of flashbacks which resemble vignettes. Vignettes are short impressionistic scenes, which purposefully paint a character, scene, or idea in a very particular light.
This technique in "A Rose for Emily" accomplishes a few things. First, it builds up the mystery and suspense surrounding Miss Emily, and makes the surprise ending all the more shocking and gruesome. Also, it serves to characterize Emily, her father, Homer Baron, and the rest of the small town against each other. The reader is not given enough information and does not have a broad enough perspective to make a well educated assessment of characters and events. Rather, it is only possible to compare characters and events to everyone else and everything else that happens in the story. This heightens the Gothic elements of the novel; the narrator's bias emphasizes that perhaps the Griersons are not the only thing in this town that is slightly eerie and not to be trusted.