What literary device is used in this quote from Romeo and Juliet?: "Well, in that hit you miss. She’ll not be hit With Cupid’s arrow. She hath Dian’s wit. And, in strong proof of chastity well armed From love’s weak childish bow, she lives uncharmed."

As well as the allusions to characters from Greek mythology, discussed in previous answers, there are also in this quotation examples of euphemism, repetition, and metaphor.
In this quotation, Romeo is replying to Benvolio, who has told Romeo that a "right fair mark, fair coz, is soonest hit." The word "hit" here is a crude euphemism meaning to have sex with. In other words, Benvolio is telling Romeo that he should have sex with Rosaline as soon as he can. Rosaline is also objectified here as merely a "mark," or a target to be hit with an arrow. The repetition of the word "hit" emphasizes the crude connotations of the word and indicates that Romeo's supposed love for Rosaline is little more than lust.
With this in mind, it seems likely that "Cupid's arrow" might be a euphemistic, phallic symbol alluding to Romeo's penis. This seems especially likely when we consider the frequency of crude and phallic imagery at the beginning of the play. Indeed, not long before, Sampson and Gregory boast about what they will do with the women of the Montague family, with Sampson insisting that, "Me they shall feel while I am able to stand: and 'tis known I am a pretty piece of flesh."
There is also a metaphor in the given quotation when Romeo says that Rosaline is "well armed" with chastity. The implication here is that Rosaline is armed with chastity as a knight might be armed with armor. The armor that Rosaline wears is, to Romeo's frustration, impenetrable. The allusion to armor also implies that for Romeo, conquering Rosaline's chastity is like a battle or a fight to be won.

As the previous Educator has noted, this quote from act 1, scene 1 is an example of an allusion. In literature, an allusion is an indirect reference to a significant person, place or idea. Shakespeare was relying on his audience having a familiarity with both Cupid and Diana to make the allusion have meaning and impact within the play.
It is also worth looking at the comparison between Rosaline and Diana in a little more detail. Diana was the Roman goddess of hunting and chastity. In making this comparison, Romeo is expressing his strong love for Rosaline. The fact that he would compare her to a goddess, a deity, is evidence of that—he has elevated her beyond the realm of a mortal woman. Romeo also believes that, like Diana, Rosaline has deliberately pledged to stay celibate. Of course, we do not know if this is really true, but, whatever the case may be, this statement is effective in highlighting Romeo's frustrations as he tries desperately to woo her.

The most obvious device is allusion, as the lines are making reference to Cupid and Diana, Cupid is the well-known flying god who shoots arrows that cause people to fall in love. Diana, on the other hand, is a virgin goddess associated with female chastity and power. Romeo is complaining about Rosaline being impervious to his charms, which brings us to a more subtle device, which is irony. The irony comes in clearly in the "hit" and "miss" pun, but plays out even more in the fact that the line reads as praise, when in fact, he is not at all happy about the girl ignoring him. It is ironic that, although he's calling her cold, he still is enamored of her, and so he sees her coldness toward him as another admirable trait. Another layer of irony resides in the fact that the shooting metaphor, derived from the idea of Cupid's arrow, also applies to Diana, since the virgin goddess is also associated with hunting.

In these lines in act 1, scene 1 Romeo is making an allusion to a well-known story from classical mythology. Most people are familiar with Cupid, the god of love, who fires his little arrows into the hearts of lovers everywhere. In the case of Rosaline, however, Cupid's arrows don't seem to be getting through. Romeo has fallen madly in love with her, and yet she won't reciprocate his feelings. Rosaline is like Diana, Greek goddess of the hunt, utterly impervious to any arrows that might start flying her way. Whereas Diana is protected from arrows by her status as an immortal, Rosaline is shielded from Cupid's arrows by her chastity. As such, she is not prepared to give herself to Romeo, however much he may desire her.