I think we can use the following sentence: I am having a headache since yesterday. correct me if I'm wrong.

Simple past is best: "My headache started yesterday," as you are implying that you still have a headache now as you say that it started but not that it finished. "I have has a headache since yesterday," (present perfect) is acceptable as it started in the past and is continuing until now. "My headache has pained me since yesterday," is grammatically correct but strange usage. We usually use the expression "pained me" to talk about a situation that causes regret or emotional upset. An example, "Your time keeping has pained me since you started working here," I am annoyed that you are always late.

Has started yesterday would be the correct one I think. The second one is ok, although people don't usually use 'pain' as a verb.