On its own, "one ought" is a clause. It doesn't serve as the subject of anything.
I will agree that reformatting the matrix clause in question, along with reintroducing what was lost through ellipsis, can expose its internal structure:
to feel them when one ought [to feel them] ,
[to feel them] in the cases in which [one ought [to feel them] ],
[to feel them] toward the people [for] whom [one ought [to feel them] ],
[to feel them] for the reasons for the sake of which [one ought [to feel them] ],
[to feel them] in the manner one ought [to feel them]
the best thing
. . . .
One ought to feel these things for the sake of some reasons. What reasons those might be is not mentioned here. To feel them for those reasons is a mean thing but also the best thing.
The sense of "sake" used here means something like "cause", "justification", or "purpose". It is in no way related to rice wine.
I am tempted to paraphrase this passage as follows:
to feel such things
at the appropriate time,
in the appropriate cases,
toward appropriate people,
for appropriate reasons
in an appropriate manner
both a mean and the best thing . . . .
The remainder of the sentence is the nonrestrictive relative clause "which is what belongs to virtue". The antecedent of the relative "which" is that thing that happens to be both mean and the best. That, in turn, shares its referent with the subject of the matrix clause.
Those three descriptions (to feel things under those conditions, something both mean and best, and that which belongs to virtue) all indicate the same referent.
In this translation, "cases", "people", "reasons" and "manner" are all common nouns in a parallel structure. They are each the object of a preposition, and they each serve as the antecedent for a following relative clause. None of them have antecedents themselves. The prepositions relate those nouns to the infinitive phrase "to feel them", which stands as the subject of the matrix clause.
The question "reasons of what?" makes no more sense than "cases of what?", "people of what?" or "manner of what?".
Perhaps you mean to ask why this translator chose "the reasons for the sake of which one ought" rather than the simpler "the reasons for which one ought".