You can't live without cell phone there (Smartphone).... It's almost impossible to find pay phone booth.
First you need compatible phone with WCDMA 2100MHz band and/or LTE 2100MHz (band 1).
They don't have 2G GSM service.
Then you need a prepaid SIM card.
Cheapest one I could find online was from EGsimcard.co.kr ($15 for 1GB of HSPA speed data with $10 voice credit).
You can order a SIM card in advance and picked it up at AREX travel center at B1F level at Incheon Airport. EG sim card comes as dual cut out (mini or micro). They can cut it to nano size when you pick up.
The SIM card has a preassigned phone number. You can also refill balance with the EGSIM card app.
When you get the sim card,
1) turn off the phone
2) insert a new SIM card and turn it on and get the notification that it is not provisioned
3) you have to turn it off and start again one more time to get the sim provisioned.
In Korea, incoming call and text are free unlike in USA.
If you have a Korean credit card or a general-purpose credit certificate, you can buy KT Prepaid sim card, they have special deals time to time. Now they are selling a $30 prepaid card (valid for 6 months) for $1 (as of March 2018). You get about 1.5GB of LTE data with $30 balance.