If you have 5 secs to answer this question then - focus on "was caught" and avoid all choices. B, D and E gone
C is awkward the moment you read it - "catching" the public’s fancy
A ic correct.
Now doing it the hard way
(A) In such works as his 1889 masterpiece, A Dash for the Timber, Frederic Remington caught the public’s fancy
>> active voice. Frederic Remington caught the fancy(B) In such works as Frederic Remington’s 1889 masterpiece, A Dash for the Timber, the public’s fancy was caught
>> passive usage "was caught"(C) Frederic Remington, catching the public’s fancy in such works as the 1889 masterpiece A Dash for the Timber, did it
>>> awkward "catching" is wrong(D) The fancy of Frederic Remington’s public was caught in his 1889 masterpiece, A Dash for the Timber, >>
passive usage "was caught". Plus "Frederic Remington’s public" is awful 
(E) The public’s fancy was caught by Frederic Remington in such works as his 1889 masterpiece, A Dash for the Timber,
>> usage of "by" makes it wrong. It become passive.