KT,
The Bagel Bar is not very close to the Fontainebleau Hotel, and depending if you mean Miami close or normal close, the Rascal House is not that close either. Will you have a car?
There's a deli in that part of Miami Beach called, I believe Arnies, around 40th and Alton that gets good reviews, but I've never been. As you go North of the hotel, there's a lot of interesting stuff, that I have mostly never tried, Russian and Argentian, although see this
post for some Argentinian discussion. If you go South from the hotel, you are smack in Miami Beach. Lotsa glamour spots, but me, I head straight to Joe's all the way at the tip. It's better (and way less crowded) at lunch.
I hope that's a good start.
Think Yiddish, Dress British - Advice of Evil Ronnie to me.