"Dian" is a reference to Diana, the Roman goddess of the hunt and the moon. She was famous for being a chilly, distant virgin, whom no man could ever touch.
The speaker's idea of Astarte is pretty different from that. He imagines her being "warmer than Dian." In this poem Astarte is a kind of sexy, emotional goddess, not cold and far away, but close to humans and full of feeling.
She rolls through an ether of sighs— She revels in a region of sighs:
Now our speaker tells us that Astarte "rolls through an ether of sighs." The "ether" is like the atmosphere, the airy regions of the sky. The speaker imagines the atmosphere being full of sighs. He also imagines that the moon goddess "revels" in these sighs.
She has seen that the tears are not dry on These cheeks, where the worm never dies,
So, up until now, our speaker has been in a crummy situation. He's been wandering around in a dark, haunted wood, muttering to his own soul. Now, all of a sudden, there's a ray of hope. He imagines that this rising star is a good sign, that the goddess has seen his pain and the wet tears on his face.
He also says something cryptic about "where the worm never dies." We think the worm is a representation of death, and the speaker's cheeks are wet because he lives in a world where the only thing that never dies is death itself.
And has come past the stars of the Lion To point us the path to the skies—
Now he imagines that the goddess has traveled up the sky, past the constellation Leo ("the stars of the lion"), to save his sorry butt. He thinks she is going to lead the way out of this nasty forest of pain and death and show him the way "to the skies."
Oh yeah, and he totally just rhymed "dry on" (line 42) with "lion." Awesome.
To the Lethean peace of the skies—
He also dreams that the skies are a place of "Lethean peace." In Greek myth, Lethe was the river of the underworld. When the souls of the dead drank from it, they would forget their past lives.
This is about as pure Poe as you get. His characters are always longing for forgetfulness and for the release of death. This speaker is no different. He wants to feel peace and to escape the pain of the world.
Come up, in despite of the Lion, To shine on us with her bright eyes— Come up through the lair of the Lion, With love in her luminous eyes."
A little repetition and rambling about the Lion here. He imagines the goddess passing through the den of the Lion (the constellation Leo), in order to help him.
Remember that all this guy has seen is a star coming up. He must be in pretty bad need of help if he clings to this fantasy so eagerly and obsesses about it like this.
We think you can really hear his pain in the last line, when he imagines the "love" in the "luminous eyes" of the goddess. He's really hurting, poor guy, and this imaginary woman is supposed to be the answer to his problems.