Winnowing out the also-rans
Garrick is mentioned in the excerpt specifically in the context of his being one of "the greatest actors of the past"—so I don't think that any doubts regarding whether modern audiences would consider Garrick's acting melodramatic or histrionic or unrealistic are conceived by the author of the excerpt as evidence of his not having been a great actor after all. If greatness in acting depends on the ability to touch the hearts of one's audience, Garrick's greatness is settled by the fact that he did have that effect on contemporaneous audiences. If no actor who has ever lived would be able to have that effect on every audience throughout history, it doesn't follow that there are no truly great actors but rather that effective acting is closely tied to popular tastes and expectations. For me, option D is not a very tempting choice.
Options B (their contemporaries would have understood famous actors") and E ("historical records can reveal personality") simply aren't the focus of the question (or "doubt") that the author of the excerpt is trying to raise—so we can dismiss them from consideration.
Option A or option C?
That leaves option A ("the stage personalities of the past would appeal on a personal level to people like the author") and option C ("the acting of famous stage personalities would appeal to us today") as viable possibilities. The author formulates two questions in the excerpt:
Should we really care for the greatest actors of the past could we have them before us? [Or s]hould we find them too different from our accent of thought, of feeling, of speech ...?
It seems to me that the crucial task in understanding what the author is asking about in the excerpt is to determine what he means by the phrase "care for the greatest actors of the past." If he means "care for the acting of the greatest actors of the past", then it follows that option C, with its focus on whether "the acting of famous stage personalities would appeal to us today" is the stronger choice.
On the other hand, if the author means "care for the actors as people," then option A (which makes no mention of the actors' acting) might arguably be the right choice.
But this GRE question's presentation of options A and C has a couple of serious shortcomings. First, it uses the term "stage personalities" in both answers—but that term is not used or defined in the excerpt or anywhere else in the question. What is a "stage personality"—an exact synonym for actor? a persona that exists on-stage but not off? a performer with a distinctive character or mannerisms, like Jimmy Durante or W.C. Fields? We don't know because the test deviser introduces the term without explaining it.
Second, the question introduces a sneaky distinction in relevant audience between option A (where the relevant audience is "people like the author") and ption C (where the relevant audience is "us today"). As Tim Romano points out, the excerpt is from an essay by Walter Pater titled "Their Majesty's Servants," reprinted in the April 1911 issue of The Mask. The GRE question deviser doesn't expect anyone to know that the excerpt is more than a hundred years old, but in a discussion of whether acting greatness would be recognized by audiences of later ages, it seems at least significant that the wording of option A links the "doubt" expressed in that option to people like the author," whereas option C links its doub to "us today."
Obviously, Walter Pater writing in 1911 can't have any well-defined doubts about the reaction to Garrick of audiences in 2015. So is this whole question an exercise in misdirection hinging on the time frame implied by "people like the author" and the time frame implied by "us today"?
I don't know. Nor am I clear on what the test deviser means by "stage personalities," although it seems to me that the farther we take that term from simple equivalence with "actors," the less connected to Pater's actual comments the GRE question becomes.
In my view, Pater is primarily interested in the acting of the great actors of the past—not in their personas or personal magnetism or whatever. For that reason, I think that option C, which focuses on the acting of the "stage personalities" and not on the "stage personalities" as things in themselves, is the better answer.
But whether you agree with that assessment or not, I hope you will agree that option A is very far from being obviously preferable to option C under any fair-minded analysis of the two options' merits. And that in turn means that this test question does a lousy job of evaluating reading comprehension, or whatever else it is supposed to be doing.