To “pull the wires” means
To manipulate or influence (a person) as if pulling the wires of a puppet; to control (a situation, organization, etc.) from behind the scenes.
Oxford English Dictionary.
A couple of citations for this sense:
I tell you Jerry isn’t given to pulling wires to get what he likes.
Beth B. Gilchrist (1917). ‘Cinderella’s Granddaughter’. In St Nicholas XLV:2 (December 1917), p. 140.
I spent two weeks pulling wires, among Senators, officers, and State Department people.
Eric Sevareid (1945). ‘Censors in the Saddle’. The Nation (14 April 1945), pp. 415–416.
So we can imagine that Monsignor Darcy is “pulling wires” to persuade people to favour Catholic causes, and “making a great mystery” of it: that is, pretending that he is doing something more spiritual than mere lobbying. Why the wires are “rusty” I am not sure. Perhaps the idea is that his connections are out of date, so that he does not achieve much in the way of influence. Fitzgerald describes him as “rather like an exiled Stuart king waiting to be called to the rule of his land”, that is, waiting for some kind of ascendancy of Catholicism, that did not happen.
The character of Darcy is based on Fitzgerald’s mentor Cyril Sigourney Fay:
Everyone who is familiar with the important influences in Scott Fitzgerald’s life knows about Monsignor Cyril Fay. Their meeting at the Newman School, the Catholic preparatory school in Hackensack, New Jersey, which Fitzgerald attended in the two years before he entered Princeton, and their friendship which developed in subsequent years until Fay’s death in 1919 have been noted and documented by every major Fitzgerald biographer.
Joan M. Allen (1974). ‘The Better Fathers: The Priests in Fitzgerald's Life’. Fitzgerald/Hemingway Annual 6. Quoted in Maggie Gordon Froehlich (2012), ‘Passionate Discretion: Fitzgerald in the Unpublished Correspondence of Sigourney Fay, Shane Leslie, and William Hemmick’, The F. Scott Fitzgerald Review 10.