If you take Karma's advice and head over to 'Little Havana" you won't find more old school setting or a better traditional cuban sandwich (since Latin American on Coral Way went byebye) than that offered at Ayestaran off of calle ocho and 27th.
The atmosphere is the real deal (been around 40 or so years and some of the waitresses too), the people eating there are real cubanos, and the food sticks in a good way. Hard to beat all around for real old time cuban. And as an added plus, you can use the nearby Domino Park (really Maximo Gomez Park) as prop if you want. Throw in some cafecito and it doesn't get much more cubano than that!
Only "Old school" cuban left on Miami Beach is Puerta Sagua at 7th and Collins.
Larios on the Beach, 820 Ocean Dr, is cuban -- may be "prettier".. they've been there since 1992, so they may qualify.
If it were me, I'd head over the water to Enriqueta's Sandwich Shop, at 186 NE 29th Street. Quite Old School and quite good.
or, head to Little Havana, where you will find much more in the way of Old School (and I don't mean Versailles).