For thee, who mindful of th' unhonour'd Dead Dost in these lines their artless tale relate; If chance, by lonely contemplation led, Some kindred spirit shall inquire thy fate,
Haply some hoary-headed swain may say, "Oft have we seen him at the peep of dawn Brushing with hasty steps the dews away To meet the sun upon the upland lawn.
Look, gang, more enjambment! The same sentence continues across a stanza break, so we'll look at both stanzas at once.
The speaker refers to himself in these lines—he's calling himself "thee."
(Fun grammar fact: most modern readers think of "thee" and "thou" as an old-fashioned, fancy-pants version of "you." But no! It's not fancy-pants at all!
"Thee" and "thou" were actually informal or more intimate versions of "you." Like French, Spanish, and many other languages that have two versions of "you," English used to have a formal and an informal way of saying "you." And it makes sense that if the poet is addressing himself, he'd use the more informal way of doing so.)
Okay, so what's our speaker actually saying to himself? He's saying that he is aware ("mindful") of the dead people who haven't been honored with lots of monuments, so he's memorializing them in these very lines of poetry.
Then the speaker wonders what would happen if some random kindred spirit, who happened to be musing on similar things (i.e., death), might ask about the speaker's fate.
He answers this question in the next stanza, and with some alliteration thrown in while he's at it ("Haply some hoary-headed" and "swain […] say")!
Probably some gray-haired ("hoary-headed") farmer guy ("swain") would say that they had often seen the speaker hurrying through the dew-covered grass to watch the sun come up on the meadow lawn.