One can disagree with Newman on many grounds. After all, Newman was writing in the nineteenth century after he had been appointed Rector of the new Roman Catholic university in Dublin. His book addresses issues specific to those circumstances, and especially to his desire to balance a general liberal education with the concept of a Roman Catholic university.
The first area where you could argue against his work would be to point out that it is based on very limited assumptions. His university lacks diversity in class, religion, race, age, and gender, assuming that all students are relatively well off white males. He makes no provision for students who are working their way through university or non-traditional students combining work and school.
Next, there is the question of whether universities exist to build a certain sort of character or simply to provide students with a limited skill set. There are arguments on both sides, and those arguments need to be grounded in the diversity of the student body. A young woman from a poor farm family simply has different issues than the son of a wealthy urban family.
Finally, there is the question of what place religion and religious values have in a university. Newman at times waffles, trying to create a via media between doctrinal indoctrination and a universal liberal education. Here, though, he seems to avoid the notion that, on the one had, in some religious traditions, it is viewed as imperative to avoid all forms of secularism, while in some other cultural traditions, any religious elements in education are problematic.