This is a very good composition, Min! I can feel that excitement you had when you got the chance to finally eat authentic Korean food that you really missed. I have to commend how you organize your thoughts. Excellent!
Here are just a few corrections:
Thus, It was long time no eat. So at that time I was missing the real one.
>>Thus, it was "long time, no eat." So, at that time, I was missing the real one.
You might not understand why I am so happy to eat Korean food which I can also eat-the home-made dishes, but the place is different!
>>You might not understand why I am so happy to eat Korean food when I can also eat home-made dishes, but the place where you eat it is different!
I was eating in China one day ago night but at that time, I was eating in Korea.
>>I was eating in China the day before. But at that time, I was eating in Korea.