I took a summer cruise and Busan was a port with a day to spend exploring. I decided not to take an arranged tour and explored on my own to check things out around the main city area which included street markets, shopping and the famous Jagalchi Fish Market. I knew very little prior to arrival and want to share my experience for others that maybe in a similar situation.
I was on the Princess Diamond and transportation shuttles ran on a regular basis to drop off and pick up in the downtown shopping district. There's nothing "at" the cruise port and you'll need to get some type of transportation to explore. Once dropped off there was plenty of things to see and do to keep you busy for day. We didn't do any temples or shrines and stuck to exploring the local street markets and shopping and walking around the area.
Highly recommend walking around the Jagalchi Fish market area. This is a huge area and you'll be amazed with the different types of seafood and creatures being offered with many places cooking right in front of you. Even if you don't eat or purchase anything, it's a site to see.
Shopping in the area includes a lot of street vendors but there is also a modern mall with the Lotte department store. Both the street vendors and mall offer a lot of food options also.
Overall a good day walking around downtown and seeing the different sites and culture. Obviously would like more than a single day but easy and a lot to explore around town if you choose to do something similar.
I made a video from the day if interested or feel free to reply if you have any questions if you're planning something similar.