Probably depends on what passport you carry. My EU passport lets me walk straight in, they stamp you passport and then you are good to go..... to the exit of the airport where you then need to take a one hour bus ride to Seoul. 

Make sure to try Korean BBQ, try to go for the mid range places, where normal people go after work to eat. 

Seoul has a thing I haven't seen anywhere else in the world (not that I've looked) indoors amusement parks. There is some old stuff to look at, but mostly like any other modern big city. It is the biggest city in the developed world. Be prepared to NOT get ripped of by the taxi drivers. 

It's easy to pass time in Seoul, ask Google for help on what to do. And of course, go to the Gangnam district..... 

Why do you need to cross a border?

Jeju is awesome, my favorite place in Asia, however at this time of year not so fun.  My friend lives there now, said its miserably cold.  If you are going to hang a bout a bit, eat bbq, black boar, eat at paris baguette, at ribs and spicy ramen, and bbq chicken.  

Jeju island is part of South Korea, once you get through immigrations and customs there you shouldn't have any problems going to Seoul. 

Seoul is my heart. You will love it there.