There are hiking and jogging trails that are on the edge of town which would be very easy to get to. The ridgeline trail system is very nice (connects to a trail going to a 2000' hill overlooking the city) and if you are a runner then Pre's trail, Amazon trail and Rexius trail. Hendricks park is renown for its rhodies put most of the bloom is done.
Eugene is a mecca for beer: The Bier Stein, 16 Tons, Falling Sky, First National, Hop Valley, McMenamins are just a few of good ones. Some of the listed brew their own, some just have a great selection of tap. There are many more.
There is an Art Museum on campus along with a anthropology museum.
Your hosts on campus should be familiar with all of the above.
ETA: It's beer week this week in Eugene but I don't know when you arrive so it may be done with.