“Is there any old school little italy restaurant left that one might visit and enjoy the classics really well made?”
NO. There is nothing good in Little Italy and hasn’t been for at least a decade. I dislike Italian American food so am not a good resource on that. But I do know, Little Italy is a gross tourist trap. Take a look at Rubirosa in NoLita or there is Carbone which is a modern-ish take on the old classics.