What does this verse mean? Do fain and use have archaic meanings here?
When first Amyntas su'd for a Kiss,
My innocent Heart was tender;
That tho‘ I push'd him away from the bliss,
My Eyes declar'd my Heart was won.
I fain an artful coyness wou'd use,
Before I the Fort did surrender:
But Love wou'd suffer no more such abuse
And soon, alas! my cheat was known:
Song: When First Amintas su'd for a kiss (1719)
Here use means ‘practice’; the sense survives in used to for a former habit.
fain is an adverb meaning, more or less, ‘by preference’.
artful may mean ‘insincere’.
Translating the line to a modern idiom: “I preferred to play hard-to-get.”
It would have been helpful if you had given more context or even got the date of the quotation close to right. It could be as late as the eighteenth century, which in my opinion, is recent enough that we should address your question.
It simply means
I want to show a false reluctance [to be kissed]
