This site contains two walks in Narni, both of which begin in Piazza Garibaldi:

1. ✴Walk I, around the area once occupied by the Roman city; and
   

2. ✴Walk II, around its medieval extension to the south.
   

The page Around Narni describes two longer trips, each of which needs a car:

1. ✴One, which starts from the railway station at Narni Scalo, covers the monuments in what is now the modern suburb of Narni.  This is possible using the bus from Piazza Garibaldi.
   

2. ✴The other is a drive to see the important Franciscan hermitage known as the Sacro Speco di Sant’ Urbano.  (You can visit Ponte Cardona en route, or walk there a a detour from Walk II).