Last weenkend,

-->Last weekend,

I met my girlfriend and then we were headed to around University called Bu-San University,

-->I met my girlfriend and then we were heading for BU-SAN University

which is quite famous in korea.

-->which is also quiet famous in Korea

Anyway, we had a small plan before we went there.

--> Anyway, we had a small plan before we headed there.

[I suggest you used headed also since in your first statement you used this word  to maintain parallelism of ideas *_^]

It was drinking together with Korean traditional drink called Dong Dong Ju, which tasts like sugary So-Ju mixed

-->We were drinking together with the Korean traditional drink which is known as  Dong Dong Ju, which taste

is like sugary So-Ju mixed with steamed rice and looks like closely white color.

Most of the koreans enjoy it, with its Cider that tastes like Sprite. We came into many areas, where it was really

cheap compared to other place around.

And we ordered a set-menu which is served with so many things.

I mean a Set-menu which we ordered comprised with side dishes we might like.

After almost 2-hours, I felt like I was going to lose my conscious but I had to regain my consciousness because

I brought my car and I still have to drive ...haha. It was great :-)