Most young male singers that you’re likely to hear at upscale cabaret rooms such as the Carlyle or 54 Below will populate their programs with well-known show tunes and the occasional pop standard. Anthony Nunziata reverses that equation: For his Café Carlyle debut, he offered mostly power ballads, channeling Elvis Presley and Nat King Cole rather than Billy Bigelow and Henry Higgins. He talked about being immersed in 1980s and ’90s pop, but most of his material came from the early days of rock — even “I Only Have Eyes for You” was more indebted to the Flamingos than Busby Berkeley.