Only D is correct in the sentence. In his old age (During that period) The other prepositions have different meanings, of course. He was an avid skier into his 70s. (He started earlier) At 50, he decided to learn English. (In his 50th year) He saved money for his old age. (So that he would have it)

B and D are the most obvious answers here, but A can also work. There's absolutely nothing wrong with B, it's just as natural as D, and they basically mean the same thing here. As for A, it's a bit different, it means "as he moves into his old age" / "during his entry into old age" / "as he started to get older", so if this is what you want to say, there's nothing wrong with A.

No preposition is wrong. Some just make more sense than others. "In" is the best answer because "old age" is an extended period of time, not a specific time. "At" is used to point to a specific time or event, so "at his old age" sounds awkward. The prepositions that work best are the ones that express containment: "in", "during", "within". "Throughout" and "into" also can express containment but they are not as good as "in" because of the words "he began". You cannot continually begin something throughout an extended period of time. You would have to change it to: "Throughout his old age he continually donated..." "Into" also does not work well with "he began". "Into" would be used to indicate action that was already in place when he became old: "Into his old age, he continued his charitable donations".