Yes, it's pronounced exactly how you think it would be. Uni is short for University, which is what kids in New Zealand refer to as their college.
Being at Auckland Uni was a complete shock to me. Just looking at it briefly, I was attending a University with 30,000 kids total, had lecture halls with up to 600-kids in attendance, and had only two papers that determined my grade for the class. No homework that was counted in, no tests, no quizzes - just two papers. That was difficult and a change, but a new way of learning.
My campus was located up what seemed like a giant hill by the end of the semester. The short climb each day had me break a sweat every time. The few of us found a shortcut somewhat around it very quickly and made it onto campus through a completely different route. Every campus building was located off of this one main road that had daily traffic traveling throughout it - which meant cross walks and even underground cut throughs.
Auckland Uni was beautiful in the spring time once the trees that lined the street started to bud with green leaves and the grass became vibrant. The streets surrounding the buildings never rested with students bustling in and out of them between classes and appointments and study sessions. Even with a very business feel, I still felt at home and comfortable within Uni - I even started to reference school as Uni.