Sure thing, you get your phone the whole time, I would leave it locked up for your mock I day or sea trials. Just keep it on silent and be attentive to the speakers (you'll be having a lot of imformative speakers) and have a great time! It's tons of fun
Seconding @Pineapples11, you'll find that there aren't a great deal of restrictions placed on what you can and can't have, so long as you aren't bringing any actual contraband along for your stay. Anything you don't feel like bringing around all day can be left by your rack, so if it's ever a concern that's always an option. I feel like I remember one of my squadmates having an Apple Watch last year, but I'm not entirely sure.
I would not bring it. I would hate for something to happen to it. Better safe than sorry. I would just wear a basic running-type watch so you will know the time, but not be worried about it. Have a great NASS.
If you wish to be a serious candidate for admission to USNA, leave the Apple watch and similar paraphernalia at home. You have been fortunate enough to be granted an opportunity many others equally deserving would like to receive, but didn't. Don't squander it communing with your watch (and by extension your phone) all week.