For libraries (again, meaning "collections of books, " not lending libraries, in the modern sense) of significant size in the year 1000, you need to look to the Islamic world: Cordoba in the west, Cairo, Damascus, and to a lesser extent, Baghdad in the east.
From
www.ismaili.net:
It was owing to his [Caliph al-Aziz's] generous patronage that the University of al-Azhar could maintain itself as a unique and distinguished seat of Islamic learning. He also created an almshouse in it for 35 men. Al-Azhar contained a huge library. The royal library of al-Aziz itself contained 200,000 rare manuscripts and an equal number of manuscripts were kept at al-Azhar. It also contained 2400 illuminated copies of Holy Koran. Later, in 436/1045 a new catalogue had been prepared in al-Azhar, listing 6500 volumes of astronomy, architecture and philosophy. When Nasir Khusaro visited Cairo, he had found 317 professors and as many as 9758 students engaged in the study of various subjects in al-Azhar. Marshall W. Baldwin writes in "A History of the Crusades" (London, 1958, p. 102) that, "The intellectual influences of Ismailism on Islam was very great indeed. During the heyday of its expansion, the poets, philosophers, theologians and scholars flocked to the Ismailite centres and produced works of a high order."
A [not-so-detailed] map of 11th century CE Cairo and its environs:
http://www.ismaili.net/histoire/photo/20misr.jpg
edit: I forgot that Tripoli also had a massive library named the Dar al-'Ilm ("World of Knowledge") that the Crusaders ransacked and razed in the early 1100's
