If we had to pick someone, or something, to compare Mrs. Pullet to, we’d go with Eyeore from Winnie the Pooh. Not to say that Mrs. Pullet is a donkey. Rather, Mrs. Pullet and Eyeore share a similar world-view. The woman is always depressed about something. She’s downright fatalistic in fact, which means that she is convinced that everything will inevitably end badly:
"Well, your husband is awkward, you know, Bessy," said Mrs. Pullet, good-naturedly ready to use her deep depression on her sister’s account as well as her own. (1.9.68)
Appropriately enough, when we are first introduced to Mrs. Pullet, she is crying. She also gets into a disagreement with Aunt Glegg, which is also a great way to introduce the character. These two sisters often act a foils to one another, and Mrs. Glegg is always scolding Mrs. Pullet for being overly emotional. The two sisters help to demonstrate differing aspects of the collective "Dodson" personality":
"Sophy," said Mrs. Glegg, unable any longer to contain her spirit of rational remonstrance, "Sophy I wonder at you, fretting and injuring your health about people as don’t belong to you [...]."
Mrs. Pullet was silent, having to finish her crying, and rather flattered than indignant at being upbraided for crying too much. It was not everybody who could afford to cry so much [...] but Mrs. Pullet had married a gentleman farmer, and had leisure and money to carry her crying and everything else to the highest pitch of respectability. (1.7.30-1)
Aunt Pullet is depicted as rather ridiculous here. And it is interesting that her depression and negative attitude are presented humorously. Mrs. Pullet is a superficial character, and she’s usually depressed for ridiculous reasons. For instance, she worries that her cousin will die and that she won’t be able to wear her new bonnet. Her concern over her bonnet is what actually sparks the concern over the cousin, not the other way around. (See the "Society and Class Quotes" for more on this scene.)
This attitude sets up a contrast with the depression and anguish that characters like Maggie suffer. Aunt Pullet’s suffering is almost a knee-jerk reaction to any and all situations, and it is also highly superficial and performative, meaning that it’s a bit of an act. The narrator notes that Mrs. Pullet is often "acting" out her grief, so to speak, because it gives her attention.
In fact, Aunt Pullet’s depression and doom-and-gloom outlook frequently manifests in medical terms. The woman is a bit of a hypochondriac, meaning that she’s convinced she, and those around her, are always sick and dying. It’s a sort of free-for-all hypochondria that coincides with her equal-opportunity predictions of doom for everyone.