To go through the quarantine procedure together, it depends at what point you meet up. If you meet up before the immigration procedure, then you can start the process together. If you want to do it this way, then find a place to wait just after getting off the plane. If you start walking down where everyone is going, you'll have to submit the health forms etc. Hang out the bathroom maybe?
The other option is to wait for your SO after you've gone through the immigration process and also have collected your bags and now are waiting at the transportation area. They will separate you according to where you are going. For example, there was a group going to the gov designated quarantine facility, another group waiting for a bus to take them to Gwangmyeong station, people waiting for pre-arranged rides, etc.
Keep in mind, meeting up before going through the immigration and quarantine procedure goes out the window if you land at different terminals. If you arrive at Terminal 1 and SO arrives at Terminal 2, then I don't know how you'll be able to meet up. The only thing I can think of is to get SO on the phone and coordinate which facility you are going before actually going there. Even that may be an issue. Maybe they end up giving you two separate rooms?
The whole process from getting off the plane to stepping outside the airport can vary from 30mins to 2 hours. No joke. There are a lot of forms you will have to fill out, temperature checks, and they will verify you have installed the app and it works before you even talk to any immigration official.
Make sure the name you enter in the app matches exactly what is printed on your passport. Oh you don't enter your middle name on the app but it is printed in your passport? You have to delete the app and download it again to start over. Not joking. There is no edit. That was the situation in June 2020. Maybe they have updated the app, but do you want to take the chance? Install the app before leaving, use the ID: CORONA after installing the app and when you are entering your info. Do all this before leaving your country.
There was a situation where a foreigner arrived but did not know the address of the gov designated quarantine facility. So he was not able to put that address on the form. I think the confusion was because he was considered a long-term visitor (has proper visas for long term stay) and in those situations they can do self-quarantine at their place in Korea (assuming they have a place already). If you are visiting on a short-term then you will be sent to the gov designated quarantine facility (at your own expense). Right now the procedure says that you will be tested at the airport and then sent to the quarantine facility. Seems only those from Europe and the US get tested at the airport.
Might be a good idea to have a snack and a bottle of water with you because the whole process can take some time and you won't have easy access to either until you get to your temporary living facility. So you would need to get this at the airport you are leaving from and not eat/drink it during the flight! Good luck!