The Platonic Academy was founded in 387 BC.
The Museum and Library of Alexandria was founded sometime around 300 BC.
It is pretty much certain that Egyptians, Sumerians and others had something along the lines of universities millenia before this.
“The world’s first university was established at Takshashilla, well known as Taxila (in India’s northwest region), in approximately 700 bce. Another large university was founded at Nalanda in Bihar circa 500 ce. According to Chinese traveler Hiuen Tsang, the Taxila campus housed 10,000 students and 2,000 professors. Students from as far away as China came to obtain a religion-based education.”
Source: Hinduism Today Magazine, March 1998
http://www.hinduismtoday.com/modules/smartsection/item.php?itemid=4717