Miss Brill doesn't really "deal with" reality—at least, not until the very end of the story. Initially, she imagines that her little fox fur is alive, even referring to it as a "Little rogue," as though it has not only living but also had a sassy, saucy personality as well. Then, when she arrives at the park, Miss Brill begins to imagine that she's playing a role in a play—that everyone in the park has such a role—and she seems not to understand that she is, like almost everyone else there, old and decrepit. It's as though she actively ignores reality, preferring her fantasy instead. She still sees herself as vital, vibrant, and important until she hears the harsh and unkind words of the young couple who sit down next to her. The boy calls her "silly" and "stupid," and the girl makes fun of her fur.
We know this has affected Miss Brill on some level because she skips her usual stop at the bakery, forgoing her piece of cake, and she begins to cry as she puts her fur away in the box (though she doesn't seem to realize it is she who is crying and not the fox). Reality seems to be breaking in to her consciousness, as Miss Brill would not cry if she was still living in her cheerful fantasy.