I went to Gakushuin University, Mejiro campus. It's pretty nice, and lively on a school day. A club or circle took the stage at lunchtime, near the cafeteria. Not a bad place to have lunch. I don't know if the campus is open to everyone, but the guard didn't check for ID when I entered. So maybe anyone can go and buy some food, have a cup of tea. I'm told that Gakushuin Schools are known for educating the royal family. Originally established in Kyoto in 1847, the university was moved thirty years later to Tokyo.
Easy to get to. Take the JR to Mejiro. When you exit the station, turn right, cross the narrow street, and you'll see the entrance.