A 51-year-old man was found dead — with his neck slashed — on an East Village street early Monday, cops said. Patrol officers discovered James Cunningham in a pool of blood on Avenue A near East 13th Street just after 1 a.m. — and rolled him over to find a large cut on his neck, authorities and police sources said. Cunningham earlier had gone into a nearby bar, drunk a seltzer and left. The bartender said he heard cops outside — and stepped out to see the victim lying dead in the street, according to the sources