Is one of these more correct?
I will apply to university next year.
I will apply to a university next year.
I go to university.
I go to a university.
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It's nice to find a usage where the Brits are ahead of the curve for once (I went to university, obviously – as did most UK undergraduates by the mid 60s).
Americans have been slower to adopt the new usage, but they're getting there...
As implied by @M. Tibbits's answer and comments thereto, I think including "a" implies slightly more "exclusivity/cachet/prestige", notwithstanding that many Americans will say "university" means the same as "college" (US speakers go to college, just as in the UK we go to university). I suggest that Brits are following this principle when they speak of going to a redbrick [university].
I believe it is more American (as per FF's answer below) to use 'a' before the common noun 'university'. But if it were replaced with a proper noun, the use of 'a' is incorrect.
I go to Cambridge. I go to a university.
I will apply to Cambridge. I will apply to a university. (only one?)
In Australia, I would find "I go to a university" a bit awkward. "I go to university" would be overwhelmingly more common, as would "I'm at university". "I go to a university" sounds like the speaker is making a point that they only attend one one, or that a university is an unusual thing. It almost sounds like they're being enigmatic about which one they are attending, perhaps because it isn't very prestigious.