She possesses neither beauty nor brain. Is this sentence grammatically correct?

Yes, it is. When using the word neither is because you want to focus on two parts of the main ideas which ones are denied. In this case first part of the sentence to be denied is the fact that she is beauty and the second one is that she is smart.

Grammar use of '' neither'': Subject + verb + neither + part one (most cases adjective) + nor + part two (most cases adjective).

It sounds correct. I'd say it is