Stanza 1 Summary

Get out the microscope, because we’re going through this poem line-by-line.

Nautilus Island's hermit

What a mouthful to begin with! Line 1, though pretty short, is kind of confusing. First of all, it helps to know that Nautilus Island is an actual place in Maine.
A hermit is someone who stays in his or her house and doesn't get out much. So Lowell is referring to a person in a particular place.
The first line is tricky because it plays with all these sea words – "nautilus," here the name of a place, is also a sea creature that leaves those familiar curved shells washed up on the beach. They're kind of like those conch shells that we press our ears to and that sound like the whooshing noise of the ocean.
"Island" has an obvious sea reference, and "hermit," though the word is being used to describe a kind of person, makes us think of hermit crabs! So right away he's playing around with the language in this poem.

heiress still lives through winter in her Spartan cottage;

OK, more adjectives to jumble things up, but if we unpack it a bit, we can find the necessary info.
So, this hermit is also an heiress (a woman who's inherited some money, probably a good amount of money).
The word "Spartan" comes from the ancient Greek soldiers of Sparta, who were rough and tough and didn't need much to get by. So here, when the word refers to the cottage it just means that the place doesn't have a ton of frills or luxuries.
Now let's also pay attention to the quieter parts of the line – "still" means the hermit heiress has probably been there a while, and "winter" takes us out of the sunny seaside picture we might have begun imagining and introduces us to colder weather.

her sheep still graze above the sea.
Her son's a bishop. Her farmer
is first selectman in our village,

This heiress owns some sheep that graze on her property.
Her son is a bishop – a noble occupation, perhaps something that she's proud of.
Her farmer (probably the guy who tends her land) holds a seat in local government. So he's mostly likely a stand-up guy too.
So all these things hint at the fact that the heiress lives a no-frills life, but she has it pretty good. Doesn't seem like she has to do a whole lot for herself.

she's in her dotage.

To say the heiress is "in her dotage" means she's in the last phase of her life. She's old, weak, and doesn't have a ton of time left.
This is an interesting shift in the description, because everything before it laid out how fortunate she is.