Basically, all big hospitals can do the job for you. (I doubt any of them would not be confidential)
If you don't have insurance, you can go directly to see a urologist or gynaecologist.
If you have insurance, you might need to register at the family medicine first, where they will transfer you to the a urologist or gynaecologist.
Here is the list of hospitals http://english.busan.go.kr/06_residents/04_01_03.jsp
I would do this on your own, don't ask anyone at your place of work for help in setting up and appointment or even a phone call. There is no such thing as confidentiality when it comes to foreigners and medical exams/checks for anything in Korea. If you involve the work place, they will find out. Korean employers are really nosey when it comes to our personal business.
There are a few clinics on Texas Street.
HAHAHAHA!!! Texas Steet;The only place where you can get an STD and get treaded for symptoms all within a stone's throw. The ultimate in one stop shopping.
I don't even trust the bottled beer there brothers. I would go to a hospital and a big one.
Hello, I work at Dongeui hospital in Yangjeong dong as an RN and there are two urologists who speak good English and prescribe to have STD test after seeing the patient.
It takes a week to get the result and it is confidential.
Please contact us at this number,010-9457-8941 and I will arrange a doctor's visit at your convenience.
Lee, hei sung, RN
How was it? Do you need an appointment for Saturdays?