There are other excellent things to see. La Spezia is not without some attractions as per this article https://www.apathtolunch.com/2014/06/la-spezia-travel-guide-top-ten.html The Lia Museum in particular is worthwhile.
You can also see nearby Portovenere easily. It's an historic town in a fantastic location. I'd suggest taking the ferry there to enjoy the Gulf of the Poets and take the bus back. The ferry leaves from the other end of the seafront promenade from the cruise terminal. For the first year you can visit Lerici and Tellaro on the other side of the Gulf in the same ferry/bus manner. Lerici is a very scenic small port town that's been featured in the NY Times and Tellaro is a perfect little town much more evocative than some of the CT villages, IMO. The new ferry service is described here: https://www.navigazionegolfodeipoeti.it/en/tour-del-golfo-goulf-tour/
Also there may well be promotional excursions available from the cruise terminal sponsored by Sarzana or Montemarcello according the photos in the local news. The latter is a movie-set pretty village and the former an historic and charming small city with several churches, a fortress, and other attractions.