I think that the first verse of the song is actually Arwen voicing the thoughts and feelings of Eowyn as she realises that Aragorn can never return her love but must fulfill his destiny as a king of men (that sounds so trite!). Eowyn's world is painfully redefining itself as she tries to accept Theoden's death and Aragorn's role in the War of the Ring; Arwen's world changes with her decision to give up passage to the Undying Lands and immortality.
The second verse applies to both women: for Arwen, the time of the Elves is passing (the leaves of Lorien are golden and falling for the last time); for Eowyn, the sun is fading as she mourns her doomed love for Aragorn (an homage to her dream--originally told to Faramir in the Houses of Healing, not to Aragorn in Meduseld--a great wavecomes, she stands upon the brink, and it is utterly dark). The last line ("I wish I could hold you closer") for Arwen is her yearning for his touch when they are separated by distance; for Eowyn it is her hopeless love for a man who could never be hers.
This is all my speculation, of course
