How does the narrator of "A Rose for Emily" by Faulkner give us insight of the initial setting compared to "Hills Like White Elephants" by...

I think you have started off on the right lines at least.
Clearly, the intial part of "A Rose for Emily" speaks a lot about Miss Emily Grierson
and her standing in the community. One approach to this question is to consider the way
that Miss Emily is presented as someone who, in a sense, was already dead or at least
not really living before her death. Note how this is indicated by the description of how
her house and its locality has declined in recent
years:

...set
on what had once been our most select street. But garages and cotton gins had encroached
and obliterated even the august names of that neighbourhood; only Miss Emily's house was
left, lifting its stubborn and coquettish decay above the cotton wagons and the gasoline
pumps--an eyesore among
eyesores.

Just as Miss
Emily's house is now an "eyesore" though it once was on the "most select street," so too
has Miss Emily herself exited life and declined in her public standing. Note how we are
told that no one had entered her house for more than ten years. She is a piece of
history, and one who hasn't really beenn living life for a long time: ever since she
killed Homer Barron and was united with him forever in his
death.