Before you can start answering this question, you need to ensure you understand what each of these terms means. Once that has been established, it's easier to think about these concepts within the context of the story.
Protagonist: A protagonist is the leading character in a literary work. The protagonist will drive the plot and is of primary importance to the work. Sometimes, it's very easy to identify the protagonist—for example, the protagonist of the Harry Potter series is Harry Potter. But in this story, there are two major characters—and, indeed, a third character, Tempy herself, is very present, even though she isn't actually there. The easiest thing would probably be to describe both Sarah Ann and Mrs. Crowe as the protagonists of the story, but your question asks for more nuance than this. We need to establish which of the woman is the protagonist and which is the antagonist.
Antagonist: the antagonist is the character who causes conflict for the protagonist, often because he or she is an "evil" character, but equally the antagonist can simply be someone who has a different understanding of the world and wants to challenge the protagonist's view.
In this story, we don't have a clean-cut protagonist vs antagonist divide; there certainly isn't a villain. What we do have are two women who come from different backgrounds, with one of them, Mrs. Crowe, rather "stingy" and of a higher social class. Sarah Ann Binson, meanwhile, is rather "anxious," and the two women are described as belonging to "opposite parties." Sometimes we judge which is the protagonist, rather the antagonist, based upon whose mind we see more insight into—we do see that "Mrs. Crowe's heart began to beat very fast," and hear some of her observations—"she had suspected before that her companion was of a weaker . . . disposition than herself." But equally we hear that "Sister Binson's quick sympathies were stirred toward this other old friend." So we cannot make a distinction based upon whose point of view we more closely understand, because the narrative does not really make a distinction between the two here.
In terms of which of these women is the protagonist, then, the "easiest" thing might be to choose Sarah Ann, as the more sweet natured of the two, casting Mrs. Crowe, the more "stingy" of the women, as the antagonist, based on the idea that an antagonist tends to be a less pleasant character. But in real literary terms, we could argue that both women are the protagonists of the story, and each is the antagonist of the other, in that they stir conflicts in each other which lead to a change in both of them.
You've also asked about character types. A "round" character is someone who has a complicated or realistic personality, as opposed to someone who is defined by one character trait alone. Both the protagonists in this story are presented as multi-faceted and interesting women, able to develop sympathy for each others' roles in life. Neither is entirely "good" or "bad." Mrs. Crowe, who may be "stingy," wishes she were more like Tempy, while Sister Binson, while thought "credulous" and "weaker" to her companion, is actually demonstrated to have a quicker wit than Mrs. Crowe thinks. Only Tempy, who is not present, is flatter as a character, being defined by her goodness.
Meanwhile, a "dynamic" character is one who changes over the course of the story. A "static" character does not. In this story, Tempy is static, by virtue of the fact that she can no longer change: she is preserved in the memories of her friends. But the other two women are certainly dynamic. The point of this story is that they are both changed by this experience, and the closing three paragraphs of the story indicate how much they have indeed shifted their viewpoint since before they spent this time in conversation with each other.