You could go to the Romanization section of this website and manually convert your address into the Roman alphabet, or you could go visit the Road Name Address information website of the Ministry of the Interior and Safety (Korean, English). If that direct link no longer works, visit www.juso.go.kr, click “English” at the top right. Now either copy-paste or type your address in the field, search, and you’ll get both the Romanization for both the new and old system, marked road name and land lot, respectively.
While people usually memorize their mailing address, it is rare for Koreans to memorize their zip code. This isn’t a problem for them as entering an address in Korean on a Korean language website requires a search and ends with the zip code is declared. This is why websites rarely display their zip code, even when showing their mailing address. To learn your zip code you can do that route on the the Road Name Address website previously mentioned, or visit the Portal Code Finder section at the Post Office’s website. If you live in an apartment, they might display the zipcode near the mailboxes or inside of an elevator on the inner doors.