Busan is what happens when a port city gets squeezed between mountains and the sea and decides to build up, down and around. During the Korean War Busan served as the wartime capital and took in hundreds of thousands of refugees. You can still read that story in the hillside villages they built and in the markets where they sold whatever they had.
It's a different kind of Korean city from Seoul. Slower, saltier and more outdoorsy. You can swim at an urban beach under skyscrapers, ride a little train along a cliff, eat raw fish upstairs from the stall that caught it and be among the tombs of an ancient kingdom an hour later. Five days is the right length because Busan is spread out: the sights sit in separate pockets along the coast, and every pocket deserves half a day rather than an hour.
The plan puts one part of the city on each day. Day one is the southern tip and the painted hillside of Gamcheon, day two the old harbor, markets and film district around Nampo-dong, day three the east coast from a sea temple to Haeundae, day four a trip to Gyeongju, and day five Gwangalli, the south shore and the bridge at night. Start on a Tuesday if you can: Gyeongju lands on a quiet Friday and day five on a Saturday, when the Gwangalli drone show runs. Busan is also an easy extension of South Korea from Seoul, a little over two hours away by KTX.