To add to the excellent responses above with specific examples, you could examine how she survived her father and his scrutiny over her suitors. She survived the disgrace of her betrothed abandoning her (and through her strong-will got revenge). Another example is how she absolutely refused to pay her property taxes. The former mayor once, because of an obligation to her father, made her exempt from property taxes. However, once that mayor is gone, the preceding ones try to get her to pay the taxes, once even venturing to her house and sitting down with her, but through her strong-will and utter refusal, they eventually cave and allow the old tradition to exist. That incident reveals a lot about Emily's character.
In addition, Miss Emily 'survives' though everything around her is changing. In the first paragraph, she is described as a "fallen monument" - she represents a way of life that no longer exists. She represents that old South, and that is part of the reason the townspeople view her that way that they do. She is a relic who survives by remaining locked up in her house untouched by outside influence - she is a "tradition, a duty, and a care" for the town.
Miss Emily can be seen as a survivor because of the many obstacles she has had to face. Her mother is never mentioned, and the lack of maternal guidance, and an overbearing father must have been hard. Miss Emily was never allowed to have a social life, as her father was always chasing away suitors whom he deemed unsuitable. The family was held in high regard, and she was always in the eye of the public. The constant scrutiny, and the loneliness seem crippling, yet she carried on. In this way, she can be seen as a survivor.