Can "me" act as a subject?

Indeed grammatically it should be "Poor little old I have to .....", but common speech has it as you have originally written - it's simply the idiom used in that fixed expression. English idioms are hard to learn!

You're wrong. The phrase 'poor little old me' is being used as the subject as though it was a third party, and not the speaker - that's why the verb form is 'has' where if the subject were 'I'. it would be 'have'.
It's like saying 'yours truly', which also means the speaker - eg yours truly has to go out again rather than I have to go out again.

Actually you have discovered one of the exceptions of the "I" subject rule. There are few of them, but this is one.

"Poor little old me has to do everything." This is actually correct. The phrase "poor little old me" is an old saying. Now I don't of its origin.

The reasoning for using "me" instead of "I" is because it has adjectives before it, i.e. poor, little, and old.

"I" is the correct form of the subject IF there are no descriptors prior to it.

This is one of those rare cases where "me" is used instead of "I".

This is a great question. It is easy to see how it can be confusing because it is a rare exception.

"little old me" is a phrase commonly used in English. Although, it does appear to be grammatically incorrect, saying "little old I" would sound incredibly strange to an English ear!