Although it dates from 1582, Edinburgh University is not Scotland's oldest (that title goes to St Andrews, founded in 1410); it is however the largest with some 13,000 students. As with many old universities, the campus sprawls over the streets and squares of the Old Town, blended with the fabric of city life.
The only thing that may indicate you are among the university's old buildings (apart from the fervent scratching of quills 'pon parchment that fills the air of course) is the odd plaque on a house or building and plenty of students either scurrying between lectures or sitting on the grass hanging out.
The Old College is the oldest surviving part, dating from 1815, but little of the university is actually open to the public, the Talbot Rice Art Gallery being an exception. With so much else to do in Edinburgh I wouldn't make it high up on your priorities, but the Edinburgh campus is a pleasant place to wander around and the university provides a couple of walking guides if you want to roam.