Uber does not exist in the "western sense" as private drivers doing ride share. When you open Uber, it will actually change the icon to UT - it works the exact same as your original Uber app, but it calls LICENSED TAXIS ONLY. Your foreign cards will work as you have them already saved in Uber app from your home country.

Also, download Kakao T. That is Korea's version of taxi hailing app ( no private ride share drives are allowed here). When you use Kakao T, when you see the "enter card details" swipe FAR LEFT you will see PAY TO DRIVER. It's hidden.