Barcelona is a beautiful city for pre and post cruise - We have been there many times. Arrange to stay several days if you can. Most people stay near Placa Catalunya and La Rambla.
We prefer the Sheraton Four Points in Poblenou, a few miles away and it’s much cheaper. I have emailed the front desk prior to our trips about specific queries and always gotten a timely response. They also have a nice breakfast buffet and the staff is wonderful. https://www.marriott.com/en-us/hotels/bcnfp-four-points-by-sheraton-barcelona-diagonal/overview/ They are at the tip of another “Rambla” though much more local and less touristy.
1. There is so much to see and experience. The HoHo buses are great for an extensive overview of the city. The mass transit (subway, tram, funicular, train) is also easy to use and they have inexpensive multi-day passes.
2. Among many sites, we have particularly enjoyed La Sagrada Familia and wandering the old Gothic Quarter.
Barcelona is an amazing city and you will enjoy whatever time you spend there.