I'm not patient, am I?

Just remember the rule for tag questions. The correct tag (am I) should be positive or negative in opposition to the main clause of the sentence, but still mirror the form of the main verb. "You (do) like chocolate, don't you?" "I am patient, am I not?" The second example is often spoken as "I am patient, aren't I?". That's an exception. (English rules always have exceptions. Don't blame me. It's not my fault.)

it's correct to say "am i?"

I think you can use the "are" form of the verb for the first person singular when you say: "I'm patient, aren't I ?"

I'm is a short version of I am. This is the 1st person singular. 'Are' is plural, though it is also used for 'you are' in the singular.