I can't remember where we ate at when we toured Case but I highly recommend fitting in the Cleveland Art museum if you can. It is free but looks like you need to reserve tickets online. My family only lives about 30 minutes away so it wasn't a big todo to go there for us so we just got something quick to eat close to campus. Neither one of my kids that toured it liked it enough to apply.
My DD lived in Oakland (CMU/Pitt) for a summer and she recommended The Yard, Stack'd or Mad Mex. Neither of my kids liked CMU enough to apply either. If you are looking for other stuff to do I recommend Phipps Conservatory. It is right by campus though it is not free. I will just say that driving into Pittsburgh freaks me out. lol You go straight from a 2 lane tunnel to a multilane bridge and you want to look around but you have to immediately know which lane you need to be in because you might be getting on a 2nd bridge and you need to make decisions quick. It sends me into a panic every time. lol Where is your hotel?
Where else are you going? For $2000 I could pick you up and be your tour guide.

(Though see above about my panicking driving in Pittsburgh and you will probably think that is a very bad idea.)