Option B is the correct answer because "in the course of time" means as time goes by, as in, along with time, so with time he became the Governor of the Province.
Option A is incorrect because "by and large" means on the whole, everything considered, which doesn't fit in to the fact that he became the governor with time.
Option C is incorrect because "at times" means sometimes, or at occasional times, which is incorrect because he doesn't become the governor at times or occasionally.
Option D is incorrect for the same reasons as option C.