You're right, "me" is incorrect there. Your revision using "I am" is correct.
"me" is the the objective case of "I". You use it when you're the object: "He gave it to me."
As we're talking about "I" and "me" I'll point out an error people frequently make, including native speakers: It's "me" even if you're not alone in the sentence. You'll commonly see "He gave James and I a lift" (even from native speakers) but it should be "He gave James and me a lift". The rule is simple: If you removed "James and", would you use "I" or "me"? Are you the subject (I) or the object (me)?