Please check my work.

I'd call "mail carrier" a noun -- it's a compound noun.

http://grammar.ccc.commnet.edu/grammar/compounds.htm

The word "mail" is a noun or an adjective in the sentence?

http://www.jiskha.com/display.cgi?id=1311122906

Your analysis is mostly correct, but there is a small error. Let me explain each part of speech in the sentence:

- "The" is indeed a definite article.
- "New" is an adjective because it describes the noun "mail."
- "Mail" is a noun because it is the object being referred to.
- "Carrier" is also a noun because it is the noun form of the verb "carry."
- "Finally" is an adverb that describes when the action happened.
- "Arrived" is a past tense verb because it shows the action of the subject.

So, your analysis is correct for all the words except for "arrived." It should be labeled as a past tense verb, not a noun. Keep up the good work!