The Arcane University in The Elder Scrolls IV: Oblivion is located in the Imperial City, specifically on the City Isle. It is situated in one of the two small towers to the east of the main White-Gold Tower, outside the main city walls in a circular area roughly the size of the innermost circle of the Imperial City, Green Emperor Way

. To find it in the game, open your map and zoom into the Imperial City. The Arcane University is on the southeastern part of the city, accessible via a bridge southeast of the Arboretum. On the map, it appears as a green fast travel icon labeled "Imperial City, The Arcane University"

. Access to the Arcane University is restricted to respected members of the Mages Guild. To enter, you must obtain recommendations from the heads of the Mages Guild in all the major cities of Cyrodiil. Once you have all the recommendations, you can speak to Raminus Polus, the Arch-Mage, inside the university to gain full access

. In summary:

• Location: Imperial City, City Isle, east of White-Gold Tower, southeast of the Arboretum
• Access: Requires recommendations from all major city Mages Guild leaders
• Entrance: Via bridge southeast of the Arboretum, speak to Arch-Mage Raminus Polus inside

This makes the Arcane University the central hub for magical learning and advancement in Oblivion's Mages Guild questline