According to the Online Etymological Dictionary, a figurative meaning of approach as ‘a way or means to address a problem’ only dates to the early years of the 20th century. Thus a 1938 Australian newspaper writer, assuming this usage may not be familiar to readers, feels compelled to frame the word in quotation marks:
One of the great differences between education to-day and in pre-war years lies in the “approach.” — The Northern Star (Lismore NSW), 2 June 1938.
Just as highways, railways, or waterways each might offer an approach to a city or grand avenues an approach to a monumental building, a figurative approach to a problem or goal most often collocates with the preposition to. A Google NGram shows that this is still the most common collocation in the Google Books English language corpus.
In the United States, the only approach to a loss of pension on account of imprisonment is the provision, by the act of August 8, 1882, that, during a pensioner's imprisonment for an offence against the law, his pension may be paid to his needy wife or minor children. — Frederick Fenning, “Comparative Pension Systems,” in Forum, May 1902, 346–348. COHA
He stretched a scrap of thin white broadcloth on the end of his finger, dipped it lightly in gasoline and worked quickly around the spot in wheel-like sweeps, getting nearer and nearer the stain, which yielded in a few minutes to gentle but steady rubbing. The gradual approach to cleaning the stain, he explained, did away with all danger of leaving the halo one often has to sponge out after a stain disappears. — Good Housekeeping, March 1903, 245–250. COHA
Two avenues of approach to a question are always open and may profitably be kept distinct: namely, the ideal, and the expedient or practical. — William Z. Ripley, “President Roosevelt's Railway Policy,” Atlantic Monthly, Oct. 1905, 486. COHA
As this new meaning took hold, the metaphoric force of motion directed toward something began to weaken, enabling the occasional use of other prepositions: for and even in:
Here, then, is a seeming approach for a definite psychological interpretation of the history of civilization. Yet the results of such a method of attack must not be overestimated. — A.L. Kroeber, Anthropology, 1923. COHA
The pattern for a new approach for teaching homemaking in the seventh and eighth grades is taken from “Homemaking – An Integrated Teaching Program” by Evelyn Herrington.
Irma Meredith Bates, “Suggested Plan for a New Approach in Teaching Home Making in the Seventh and Eighth Grades of the Junior High School at Freeport, Long Island,” Cornell University, 1937. Typescript.
Failure to seek a common approach for deciding when subversion has occurred, and what should be done about it, would leave the U.S. in the position of having to do both itself in case of future crises like the Dominican one. — M.J. Rossant, “Dilemma,” New York Times Editorial, 23 May 1965. COHA
An approach for rather than to necessarily makes the goal, and especially the purpose, of the approach more topical than how to get there:
Another approach for getting at total medical care expenditures for a sample of consumers is to estimate expenditures according to the units for service consumed. — Vital and Health Statistics, Series 4, No. 17 (9/1975).
The new regime in Uganda had not officially approached Libya for the return of deposed dictator Mr Idi Amin, Uganda's Foreign Minister, Mr Allimadi, said in Canberra yesterday. — Canberra Times, 29 Nov. 1979.
That an approach for Idi Amin’s return also appears in a headline brings up an interesting usage feature of approach for: it is vastly more likely to appear in a headline or title than in the body of an article. Even the approach for teaching homemaking in a Long Island junior high, despite being a complete sentence, is actually a chapter title.
A Modified Multhopp Approach for Predicting Lifting Pressures and Camber Shape for Composite Planforms in Subsonic Flow — John E. Lamar, NASA Langley Research Center, 1968.
'Damage control': an approach for improved survival in exsanguinating penetrating abdominal injury. — J Trauma 35,3 (Sept.1993), 375-82.
SuperStar: a knowledge-based approach for identifying interaction sites in proteins. — J Mol Biol. 1999 Jun 18;289(4):1093-108.
A systematic approach for evaluating the quality of experimental toxicological and ecotoxicological data. — Regul Toxicol Pharmacol. 25,1 (Feb. 1997),1-5.
A systematic approach for evaluating and scoring human data. — Regul Toxicol Pharmacol. 66,2 (July 2013), 241-7.
Shuffled complex evolution approach for effective and efficient global minimization. — Journal of Optimization Theory and Applications 76, 3 (March 1993), 501–521
The difference, then, between an approach to and an approach for is less lexical than it is topical, but most of all the types of texts approach for is usually found: in headlines and titles.