Hi, I'm a recent University graduate considering teaching in South Korea.

I like the idea of teaching people and would try to do a good job but I'm worried that my
shyness would make me a bad teacher. I get nervous speaking in front of people, (I hated
giving presentations in University), and am worried that my pupils will just make fun of
me.

They will. South Koreans are arrogant and rude. If you find a place where this is not the case it is the exception and you should consider yourself unusually lucky.

Also I've heard children in hagwons can be very rude/disrespectful towards foreign teachers
so I'm very worried I'll have problems with being able to discipline leading to conflict
with my boss or the pupils parents.

They most likely will be and you most likely will.

Would it be so bad that a shy person would find it too difficult?

Yes. I recommend going somewhere else. Korea is a strange society at best and deliberately abusive at worst. It is a bad time now to go there. Try some place else. Even Russians are more honest and warmer and better all around people. Don't be lured in by the promise of easy money in South Korea. It's not easy and it's also not much money anymore.