Willibaldsburg is a must see for anyone visiting Eichstätt. Of course, you can "see" it from just about anywhere in the city, but it's worth the hike or the drive up here.
There is a nice restaurant at the top where you can grab a beer and enjoy a wonderful panoramic view of the surrounding cities from their beer garden.
There is also the beautiful botanic garden "Hortus Eystettensis" which has another great panoramic view of the city. Keep your camera handy.
The museum (Jura Museum with Archaeopteryx) looks great, but it is only in German, so I skipped that.