Is there a word, or phrase, or a better way to say that it is "easy to get diarrhea” for you？
(Some people can drink two-weeks-old milk and be fine, while others get diarrhea drinking 10-days-old milk, etc.)
This is an interesting question, Mikichan, because a complete answer probably requires more than one sentence. People use different registers of language to talk about something like diarrhea.
In my part of the world, it's common for people to use "the runs" when they talk about diarrhea in everyday conversation: Something gave me the runs. = Something I ate gave me diarrhea.Everything gives me the runs = I get diarrhea easily.
If you are talking to people who might be offended by frank talk about bowel movements, you can use a euphemism like "sensitive stomach" to refer to the problem indirectly: I have a sensitive stomach = I have problems digesting my food/I suffer from problems from diarrhea/indigestion/etc.
Paul Q and Marc B gave you fine answers for a frank yet inoffensive way to mention the problem. Their suggestions sound very much like something a parent might tell a physician about a child.
'Prone to (getting/having/catching)' something is another option in common use.
'She's prone to nosebleeds'
'My son's prone to diarrhea, if he drinks cow's milk'
'I wrap up well in the winter because I'm prone to catching bad colds'
'Children are prone to get/getting infectious diseases when they start nursery'
Apart from slang or highly colloquial terms, 'diarrhea' is in common use and I recommend using it and not any of the euphemisms such as an upset stomach, if that is part of your question.
By the way, I find your example of drinking old milk an extremely odd one, probably because who in their right mind would be drinking it. I don't think we would usually talk about being 'prone' to food poisoning. If we were talking about that sort of thing we would use 'susceptible to'
'The elderly and the very young are more susceptible to dehydration in very hot weather'