heiress still lives through winter in her Spartan cottage; (line 2)
This is a mixed up line, because heiress means she's inherited something, but Spartan means simple and spare. So she's trying to keep up the appearance of being "old money" but probably doesn't have that much.
Her son's a bishop. Her farmer is first selectman in our village, (lines 3-4)
What the old woman does seem to have a lot of is pride. Her son's profession in the church and her farmer's small political office are points of pride for the woman, elevating her above "common folk."
the hierarchic privacy of Queen Victoria's century, (lines 8-9)
Again, it seems the woman considers herself the queen of this small town, and doesn't want anyone to challenge her.