"We can not drink cold drinks when that thing comes."
"That thing" refers to her period, as known as Little Auntie (小阿姨). She also refrains from drinking cold drinks during the "cold months" of October through May.
"This time my period lasting longer because I ate Yaodun Paigu."
Yaodun Paigu (藥燉排骨) is an herbal stew of pork ribs and Chinese medicine.