Please check my work.

As Writeacher told you, you're correct -- except that "mail carrier" is usually considered a compound noun. It's possible, though, that your text considers "mail" an adjective describing "carrier.

I need to know what part of speech is the word "mail" in the text.

"Mail" is part of the noun.

We can consider "mail carrier" a compound noun, similar to:

bus stop
washing machine
swimming pool

http://www.englishclub.com/grammar/nouns-compound.htm

Thank u.

Is "mail" a noun or an adjective in the sentence?

mail+carrier = a compound noun

You don't list the words separately.

Okay.

This will be my finally draft.

The (definite article) new (adjective) mail (noun) carrier (noun) finally (adverb) arrived (past tense verb.

Thank u both Ms Sue & Writeacher.

No!!!!

As Writeacher and I have tried to explain, you can't separate mail and carrier.

"Mail carrier" is one noun.

Please discuss this with your teacher.

Also show him/her the websites we've posted about compound nouns.

I got it.

The (definite article) new (adjective) mail carier (compound noun) finally (adverb) arrived (past tense verb).

Thank u again.

Yay!!!!!!!

You've got it!! :-)

You're welcome.