A nearly-50-year-old institution, the Harlem School of the Arts, shuttered suddenly on Friday and may close for good if it can’t find can an immediate source of cash. The financial situation at the school, located at 645 Saint Nicholas Avenue, has grown so dire that city officials say they must drum up $500,000 just to stay open for the rest of the year. While teachers and students say that mismanagement is to blame for the closure, school officials, including Christopher Paci, chairman of the board, said that the financial downturn is the main culprit. “As the economy turned and a number of our institutional donors cut back on their giving, our revenue from fundraising shrank dramatically as our expenses remained the same,” Paci said. The school has been a pillar of the community for generations, according to neighborhood residents, educating students in music, dance, theater and the visual arts, while offering partial or full scholarships to students.