Thomas J. Watson Library at the Metropolitan Museum of Art
Monday, Tuesday, Thursday, Friday, 10:00am-5:00pm, Saturdays
10:00am-4:00pm
https://www.metmuseum.org/art/libraries-and-research-centers/thomas-j-watson-library
To visit the Library you must request a pass at the Information desk in the Main Hall or the Admissions desk downstairs. After which you can enter the Museum freely. The Library is closed on Sundays, which kinda limits you if you were hoping to visit the Met on a Sunday but if you're looking forward to visiting the Met on a Sunday your mind is not inquisitive, it's warped.
Should you decide to use the Library on a regular basis you should apply for a Library card, which will give you free admission to the Museum on a regular basis. Of course, if on your way to the Library you should decide to refresh your heart and brain with a detour through the galleries, But it would be wrong. (Who said that? I forget.)