As the last poster already said, you are basically doing an arrival and then get back in. Unless you are flying private, all commercial flights are from the same terminal. It is set up as a point to point airport as there is no airline using it as an international hub, thus there is no Transit area.

I am guessing you will only find out about the need to collect the luggage when you check-in at US departure and if they indeed tag it to final destination or to NAP only.

Capodichino is indeed on the smaller side but not that small. Food options are actually pretty good, both at arrivals landside and departure airside.

On arrival, passing passport checks and picking up your luggage (if needed), you follow the exit signs and after the automatic door, you can turn straight left and go towards departure or right to go to a nice Pizzeria (wood burning oven)/Restaurant.

if opting to go towards departure, and the FlyDubai counters are still closed, you may want to go upstairs and grab something from one of the food outlets or do some shopping. If however, your luggage is checked through, you may want to go through security and spend the time airside. “Marenna Assaggi” would be my choice, but there are also a great pastry shop (Bellavia) and Mozzarella outlet which I often pick stuff from. Having access to the VIP lounge, although not a great space, it has a great choice of light food Buffett (my personal favourite is the fresh Mozzarella Di Bufala with some Taralli).