'apply on' or 'apply to'

Solution 1:

I'm looking forward to reading the interpretations of others here. Straight up, I would say that it's:

We apply this novel approach to the naninani corpus.

Not to say that the other is incorrect, but because - and it's just opinion here - apply to is pretty much a phrasal verb.

Looking only at the preposition to, however, it indicates movement towards a destination. It's not entirely analogous to actually reaching that destination. Well, not without further information. You can apply to university, but that doesn't mean you get accepted.

The preposition on conveys something different, it implies a connection has been made, whether that be physical or abstract. Take waxing a car; you can apply wax to a car, no problem, but really you're applying wax on the car. Or onto the car, if you prefer.

Coming back to your question, the choice will be dependent on your focus. I still feel that

We apply this novel approach to the naninani corpus.

is more fitting. It covers a broader interpretation, and it's less of a sticking point for readers. If you want to focus more on the results of the application you could consider using

We apply this novel approach on the naninani corpus.

and then get stuck into those results.