Although the adjacent Augustus Gardens is open and worth a visit for its wonderful views of the Faraglioni rocks on the one side and the Bay of Marina Piccola on the other, the Via Krupp pathway (with its drop of 300 feet) has been closed for decades due to a danger of falling rocks. You can still look down onto the Via Krupp from up above. Perhaps it's a blessing that the Via Krupp is closed... if you went down, you would then have to get back up!
In addition to the Augustus Gardens, you can visit the Charterhouse of St. Giacomo, built in 1371, one of the oldest structures on Capri. The former monastery and cloisters today houses a High School, the Diefenbach Museum and, during the summer, functions as venue for concerts and cultural events.
From Capri's Piazzetta: It is about 1/4 mile walk along Via Vittorio Emanuele, Via Federico Serena, and, finally, Via Matteotti.
Click HERE FOR WALKING MAP
Entry: 1 Euro.
Open from 1 March to 15 November - 9AM–7:30PM
Free for children under 11
Entry for both the Augustus Garden and Charterhouse of St. Giacomo - 3 Euro
The best times to visit are early morning and late afternoon, when the gardens are less crowded.