Finkelstein studied in New York initially in art classes sponsored under the Works Progress Administration, 1937-40, and then at the Cooper Union, 1940-42. Following four years of military service during World War II he returned in 1946 for another year of study at the Cooper Union, and then worked at the Art Students League, 1947-49; and the Brooklyn Museum school, 1949-50. His first of many one-man exhibitions was presented by the Pyramid Gallery, New York, in 1950.
His honors and awards include two Fulbright Fellowships supported two years' painting in Italy, 1956-58, and a year of teaching art in Brazil, 1981. A frequent lecturer and contributor to the more prominent art journals, he has received the Magazine of Art Award for Critical Writing.
While continuing an active role as a painter, Finkelstein is an especially devoted and respected teacher. He has served on the faculties of the Brooklyn Museum school, 1950-56; the Philadelphia College of Art, 1958-62; Yale University, New Haven, Connecticut, 1962-64, and in the Yale summer art school regularly since 1962; Brandeis University, Waltham, Massachusetts, visiting professor, 1977; and since 1964 at Queens College of the City University of New York. In 1979 he received the College Art Association of America's Award for Distinguished Teaching of Art. He also frequently serves in an advisory capacity on art education related projects around the country; among such special appointments have been: consultant in curriculum and planning, University of California at San Diego, 1965; review committee, Carpenter Center for the Visual Arts, Harvard University, Cambridge, Massachusetts, 1973; review panel, Council for the International Exchange of Scholars, 1978-81; consultant, New York City Department of Cultural Affairs, and International Council for Educational Development, both in 1979.
In his painting, Finkelstein is a landscapist, whose work reflects his admiration of the French schools of Pointillism, Fauvism, and the late work of Monet. In 1989 he was awarded an Altman prize in the Academy's annual exhibition. He is represented by the Ingber Gallery, New York.