Nothing is open that early. Yes, people go to work, but Korea is not a busy country until literally 11 AM. That's when malls, most shops, etc. open.

So if you land at 5, enter Korea, get to Seoul around 8 AM. The most you can do is go to a cafe before tourist sites/museums open. Plan to spend your day in the Euljiro/Gyeongbokgung/Samcheon-dong area. Gwangjang Market, Namdaemun Market, Seoul Tower, etc. The main tourist area but also the original downtown.

If your flight is at 6:45, you need to be at the airport at 4:45, which means you need to leave Seoul around 3-3:30 depending on how you will go to the airport and which terminal.