Well at home I am using this wireless router to connect to the internet, but can I just bring the router and set it up in university room and continue using the same internet there?
If I were you I'd bring a cat-5 (ethernet) cable for a wired connection when you're not wireless. From my experience, wireless at universities can be unreliable and often gets terribly slow during peak hours.
I would not recommend cat-5 but either cat-5e, cat-6

