I found two mistakes in this conversation. Doesn't "I'll bring it" have to be corrected to "I'll bring one" as there was no specific cooler designated among them? And "cooler" definitely is wrong without "a". If you find some other mistakes, please let me know along with the answers.
Yuna : Tomorrow is the picnic day!
Minjun: I know. Tomorrow will be hot. I'll wear shorts.
Yuna : Maybe we can bring cooler to drink cold water
Minjun : OK! I'll bring it. Let's meet at 8:00 am
Yuna : It's so early! Can you give me a wake-up call?
Yes, it's wrong without "a." It could be "my" or "our" as well. More natural would be "Maybe we can bring a cooler of ice water."
And yes, it is more natural to say "I'll bring one" in this case.