Does "Gentleman that I am" mean "I'm a gentleman"? I'm not familiar with the structure of the sentence.
Yes, "gentleman that I am" does mean "I'm a gentleman." He may be using this sentence because he is emphasizing that he is a gentleman; that he is so noble, that he needs to help a woman in need. "Gentleman" is the noun, "that" is the definitive article that shows that the gentleman is Flynn Rider, and "I am" is the independent clause.