I am not your mission president. I am not an apostle. You really have to look to those guys to figure out what you should be doing since they are your priesthood line of authority.
However, there are some tips that apply especially to Koreans.
Koreans love their culture just like you love yours. Be open to their culture! There are a lot of things they do that are different, but still good. Adapt and enjoy.
Remember: They are Human Too
Because of the language and culture barrier, you may start to think that they don't think like you or they don't care about the things you care about. This is absurd thinking. They show their concern differently, but ultimately, they care about the same things.
Eat, eat, eat!
My policy was "eat first, ask questions later." You really can't think about what you are eating, too much. Just try it and see if it tastes good. If it does, ask what it is and ask for more. A lot of food in Korea is stuff you simply cannot get outside of that country. Even the Japanese and Chinese are jealous of the variety and flavor on that peninsula.
But more importantly: eating together builds a bond of friendship and trust. Exploit that. You'll find you'll trust your friends more as you eat with them, and they'll trust you more.
Don't Stop Talking
Never stay quiet. You weren't sent far away to listen and watch. You were sent to speak. Start speaking. Never stop speaking. Don't worry about culture norms when it comes to talking to people and introducing yourself. The worst people can do is ignore you. (Well, they can do worse, but they are not likely to.)
Don't Stop Reading
When you have no one to talk to, then read in Korean. Don't worry that you don't understand. Your brain will put it all together eventually. The more often you expose yourself to the patterns, the more likely your brain will identify them.
Think of it this way. Let's say you want to learn all the streets in Paris and how they fit together. Either you can try to memorize the streets one by one, or you can watch people as they walk the streets to get from one place to another. If you watch how people get around, you'll learn how people really use the streets much quicker. Get the "feel" of the language by experiencing it as much as possible. Trust your brain! That's what it does---make sense out of crazy things.
"What is that you are carrying?" "What job do you do?" "What is your favorite food?" Learn how to talk, and while you talk, learn how to get others to talk. Ask questions. The dumbest things are fascinating to foreigners. Koreans love explaining their culture and life to people, especially the things that make them unique in the world.
If you have a hard time with this, think of questions you want to ask, then translate them to Korean. Then practice on someone you trust to help you get it right.
If you're asking questions you're genuinely interested in the answers of, you're going to get a better response.
If someone says they already have the gospel in their lives, ask them what Jesus and the message of salvation mean to them. Get them to share details about them as much as possible.
Koreans will say things like, "Unlike many other countries, Korea has four distinct seasons." This is, obviously, an absurd thing to say, but it's stupid of you to quibble. Just nod your head, say, "Amazing." or "Really?" and move along.
The worst you might get is, "America stinks because..." Some of the things are true, but a lot of it is outright falsehoods. Again, you are not an ambassador for the United States, you are representing the church. So nod and say, "that's interesting. But I wanted to talk about..." A really great phrase is, "If America is really guilty of that, then it's a bad thing." (미국은 진짜 저렀으면, 나쁩니다.)
Some people will try to argue with you about the gospel. Again, you don't need to argue. Switch the conversation to find common ground, and then build on that. With Christians, say something like, "We believe in Christ, the Son of God," and then, "We believe Peter and Paul spoke the words of Christ by revelation." See where this is going?
Contention has no place in missionary work. If you can't do your job without contending, then move on to the next person. I guarantee you will get 0 baptisms with contention.
Truly lose yourself. Lose your American-ness. Lose your you-ness. Devote yourself, 100%, to being a Korean Missionary. Become, in all senses of the word, a living example of what Christ would be if he were born Korean, as much as possible. Ammon gave up himself and became the king's servant. You should similarly become devoted to the Korean people.
When you do this, you'll find yourself loving them more deeply than you've ever loved anything or anyone. You will feel renewed energy and enthusiasm for your work. Every word you speak will be carried to the hearts of the people you speak to with sincerity and compassion. They will reciprocate. Even though you can't possibly master the language in the short time you have, the conviction and sincerity of your broken Korean will be enough to baptize a lot of people.