I would've said the same thing about my own level, but two schools I applied to placed me firmly in the advanced category. The impression that I've gotten is that it's all about having laid a foundation. Though we have a LOT left to learn, I think the expectation is that it'll come naturally and "advanced" is not a measure of fluency but a level of learning, or knowledge of grammar. Either way, it sounds like you'll be perfectly fine.
I feel the same way about Al-Kitab. If you want to review grammar hardcore, I'd suggest A New Arabic Grammar by Haywood and Nahmad. It's old-school pedagogy that focuses on the little details of grammar. Congrats on your acceptance to NYU!