One of my absolute favourite things I did in Busan was go on a hike up Geumjeongsanseong Fortress..
I don't know how Reddit feels about promoting.. But I found the tour in Viator (Hiking and tasting Rice wine in the Mountain of Busan) The guide was so friendly and knowledgeable and was patient and kind when I needed to take a break or slow the pace.. It wasn't a private tour but I lucked out and was the only one booked for that day..
Then on the way down we stopped in a mountain village and had Pajeon (Shallot Pancake) and Makgeolli (Rice wine)..
It was such a fantastic experience and so great to see a different part of Busan / different part of South Korea.. Definitely a highlight of my trip and something I will remember for a long time :)
10/10 would recommend