Her guitar is a Fender Japan '57 Reissue Stratocaster in Surf Green. Interestingly, this is about as far away from Yui's guitar as she could get, as Fender and Gibson are the two most well-known guitar manufacturers, thus each of their most well known models, the Stratocaster and Les Paul respectively, are often directly compared and seen as rivals. The sisters' choice in color is also essentially opposite, one being red/orange, the other greenish-blue. Despite Fender Japan's site not listing Surf Green as a color option for this model, it is available. Fender Japan's page for the ST57.
Ui's name, depressingly enough, means "grief" and "worry". However, one interpretation is that Ui is basically someone who "worries" about people (namely, Yui), and is "kind" to "people" around her (If you put the radical for person next to the kanji for her name, you get 優, the kanji for "kindness").
Whenever someone tries to point out Yui's negative qualities, Ui can only see them as either being positive or cute; she also tends to get a little defensive.
Ui sometimes helps Yui with song lyrics for the Light Music Club. She helped co-write Gohan wa Okazu with Yui, and Ui herself is the main inspiration for the song U&I, written by Yui after she comes to the realization that she can't rely on Ui forever.