A shy, religious-minded publisher who had married a dukes daughter

Meaning Analysis:

Harold Maximillan was a shy, religious-minded publisher who had married a duke’s daughter. Harold Maximillan's rise to the position of Prime Minister in 1957 surprised many people, but since the 1930s, Churchill had been extolling Macmillan’s courage.

Error Analysis & Process of Elimination:

"A shy, religious-minded publisher who had married a duke’s daughter" has no subject and is therefore a clause that is modifying the noun "Harold Maximillan" (as it is giving information about Harold...."being shy, etc.")
In A, the clause incorrectly modifies "Harold Maximillan's rise" instead of "Harold Maximillan"
Hence, eliminate A & B.
C & E also modify "Harold's becoming Prime Minister" and "the position of Prime Minister", instead of "Harold Maximillan"
Hence, eliminate C & E.

D is the correct choice as the preceding clause modifies the noun "Harold Maximillan" correctly.

All one needs to know is that "The noun which is modified by a clause should be placed immediately after the clause (after the comma)".
Once this rule is known, it's a clear 4:1 split in the answer choices here.