Clearly "mattina" and "pomeriggio" are nouns. For example "la mattina ha l'oro in bocca" uses "mattina" as subject of the phrase.
However, when I say "Ci vediamo domani mattina" is "mattina" an adverb like "domani"? If not, isn't it missing a preposition (for example "domani di mattina")?
Also, is it correct to say "Ci vediamo pomeriggio?" (or "mattina")? If not, why not?
In the phrase “Ci vediamo domani mattina”, “domani mattina” as a whole is an adverbial phrase (locuzione avverbiale), and as such it does not need a preposition.
You can't say “Ci vediamo pomeriggio (mattina)”, because you either need to specify which day (like “oggi pomeriggio” or “domani mattina”) or use a preposition if the context is clear (“quando arrivi domani?” “arrivo nel pomeriggio”).
You can also use an indeterminate article if you want to express that the day you are referring to is (still) unknown (“ti telefono una mattina”).
Edit:
You can say “domani di mattina”: while maybe less elegant (probably subjective), it puts a slight emphasis on the fact that it's in the morning compared to “domani mattina”.
I've investigated my own question a bit and I believe I can answer it fully now.
Mattino, mattina and pomeriggio are nouns and not adverbs. They don't have a double noun/adverb form like oggi or domani (I can say "Ci vediamo domani", but also "penso al domani").
When I say "Ci vediamo di mattina" or "Ci vediamo domani pomeriggio", "di mattina" and "domani pomeriggio" form a "complemento avverbiale di tempo". However this is logical analysis which is not what the question is about.
Grammatically speaking, "mattina" and "pomeriggio" remain nouns, and that can be seen in the (relatively archaic) usage of the words in the form "ci vediamo la mattina" or "Ci siamo incontrati un pomeriggio".
Nouns take articles, adverbs don't. For the same reason "ci vediamo pomeriggio" is incorrect, as the lack of an article identifies "pomeriggio" as an adverb, but there is no such form of the word.
