To some extent, the answer to your question depends upon where you live. Different areas of the world use slightly different grammatical rules for the English language. Therefore, the reaction to your sentence would depend upon where you were using it.
In the United States, you would add an article such as "the" or "a" before the noun "university," to make your corrected sentence something following the pattern of, "I am enrolling in the university next year." Note that you would use "a" since the vowel university starts with sounds like the y sound..
In the United Kingdom, the sentence as you presented it would be acceptable. British English grammar does not use articles in that type of sentence construction.
In the United States it is much more common to say, "I am enrolling in college next year." If you have a specific university in mind, then it would be correct to say, "I am enrolling at Harvard or Columbia or Berkeley or wherever next year." And in that case you would say "at" rather than "in." But I don't think any American would say I am enrolling in university or enrolling in a university. "College" is a sort of generic term that includes colleges and universities. A lot of people will say something like, "I'm going to college after I graduate," when they intend to go to a university. Come to think of it, I don't believe it is customary to use the word "enrolling" unless you have a specific institution in mind. "Enrolling" means going into the administration building, finding the right window or office, and signing some papers and probably paying some money--going through a formal procedure.
Wow, thank you for the detailed and informative answer.