先に泣かれちゃったら　ワタシはつよがるしかないよ. I believe 先に here means "first", i.e, the speaker burst in tears before the other person. "ちゃったら" (しまったら) adds the sense of ending up in an unwanted situation against one's will. Without it, the sense of "being helplessly swayed by emotion" won't be as clear. And I think 泣かれちゃったら means "if I end up crying first" rather than "because ...". So I would say it means: If I end up showing tears first, I would have no choice but to act tough.