'Old school' as 'traditional'
Instances in which "old school" doesn't literally refer to a school go back to the mid-1700s. One example is from Thomas Potter, "The Reply of the Country Gentleman to the Answer of His Military Arguments" (1758):
And how could that cruel Wind from the Bay be kept from forcing such a a Sea over the Sand-bank before Fouras, as would prevent Boats from passing the necessary Distance of three Miles even to a Transport: Yet this Account of Mr. Knowles's you believed; I should have said the Generals believed it, they reasoned upon it, and they concluded upon it. So that any Attempt to land nyy where in the Neighbourhood of Rochefort was against every military Rule of the old School for the Purposes of a Coup de Main, unless one could have fix'd the Weather-cock to particular Points.
...But this additional Artillery ["a larger Quantity of Battering Cannon"] was denied, the two old Battalions were denied, and what was still infinitely worse, the obstinate Minister, contrary to every Principle of the old School, persisted in his Opinion that the Secret was not to be risqued for the sake of more minute Intelligence: Ignorant Man! Who could think it a sufficient Foundation for an Attack that a Town was known to be without Fortifications, and without any considerable Garrison, when he might have learnt by sending Spies, and making Enquiries at the Risque only of the Secret, the exact and precise Number of Guards which mounted every Night, and the very hours when they went the Rounds to relieve them: Had these Circumstances, which the Books of our Generals declare to be necessary, been known, the rest of the Enterprize would have been easy.
Here, "old School" is evidently an allusion to the teachings on military tactics laid out in "the Books of our Generals"—but the term is evidently at least somewhat figurative.
From Edmund Burke, "An Appeal from the New to the Old Whigs, in Consequence of Some Late Discussions in Parliament, Relative to the Reflections on the French Revolution (1791):
In proportion, therefore, as the government of France was shaking by external conspiracies, and trembling for its existence, it became of course more subject to internal agitation by the revolts of its own subjects. Had it not therefore been for our unhappy interference, royalists of the old school, and royalists of the monarchical revolution, bending before the storm of national opinion, and seeing no great standard hoisted for their protection, would have really or seemingly acquiesced in the new order of things; ...
From a review of A Treatise on Practical Navigation and Seamanship in The Critical Review, Or, Annals of Literature (October 1798):
A Seaman of the old school gives us the result of the experience of above fifty years. He tells his plain tale without much regard to accuracy of diction; but he amply compensates for the defects in his style by the soundness of his judgment, the freedom of his remarks, the naïveté of his manner, and the strength of his piety.
And from J. Vage, "Observations upon Inveterate Ulcers," in The Medical and Physical Journal (November 1799):
The result of my trial, both in respect of theory and practice, I shall communicate as briefly as I can for your useful publication. In this, however, I must beg leave to speak agreeably to the doctrines of the old school; for, though the present rage of applying the modern chemical discoveries to the practice of physic is highly commendable, yet as a warm controversy subsists between men of eminent abilities, not only about all the applications which have been made of these discoveries, but even about the discoveries themselves, it warrants me to decline them, until indubitable facts, and cool impartial testimonies, shall have decided the business on one side or other.
Many additional instances of "old school" in the sense of "traditional" appear in the first three decades of the nineteenth century.
'Old school' versus 'new school'
One very early occurrence in which "old school" is explicitly contrasted with "new school" is from a letter of Nathan Beman written in September 1831, reprinted as "Review and Vindication," in The Calvinistic Magazine (December 1831):
This "difference in theological views and attachments," as stated by the Editor [of The Christian Advocate], has led him to the classification of the members into "The Old School and the New School Presbyterians." The Old School are characterized in the following manner. "In the first class we include those who put the same construction on our Confession of Faith, Chatechisms, and Form of Government, that was put on those Formularies when the Constitution of our church was adopted, an for several years afterwards." ...
From this discussion it would seem that the majority and minority in this last Assembly, were not formed on the principle of doctrinal distinction, but on the principle of ecclesiastical order. The terms Old School and New School, more properly mark the difference in theological sentiment, and are employed somewhat incorrectly by this writer, where the terms minority and majority, or High Church and Low Church parties, would have been more simple and appropriate, and much less liable to mislead the reader.
Similarly, from The Case of the General Assembly of the Presbyterian Church in the United States of America, Before the Supreme Court of the Commonwealth of Pennsylvania (1839):
Mr. Meredith.—I understand you to say that you are Old School men?
Witness [Mr. Auchincloss].—We are of the Old School, as the New School party call us.
Mr. Meredith.—Do you know by whom those terms were first used?
Witness.—They were first used by the New School party in the General Assembly of 1831.
Mr. Meredith.—Do you know who first used that term?
Witness.—I do not know, but believe it first came from the neighborhood of my respected friend here (Dr. Peters.) The term has been used for a long time.
Another repeated occurrence of "old school" from the same era involves competing approaches to regimen's of medical treatment, where two competing schools of thought regarding scientific medical reform—the Thomsonians and the Worthingtonians—were fighting each other for the mantle of progressive medicine (involving homeopathic therapies based on botanicals and steam), while characterizing mainstream medical practice as "old school." The Western Medical Reformer: A Monthly Journal of Medical and Chirurgical Science: Edited and Published by the Medical Professors of Worthington College, volumes 1–3 (1836–1838) includes many references to "old school physician" and "the old school fraternity" of practice based on "mercury and the lancet" [the latter referring to bleeding as a treatment for various physical infirmities]. For example, from "A Case in Regular Practice," in The Western Medical Reformer (November 1838):
The exclamation "I am an old school physician," at once puts a stop to to all inquiry into their qualifications or proceedings; and should a patient expire under their treatment, the friends are consoled by them with the assurance that every thing that could be done, was done; his days were numbered and finished, and he died not with empirical treatment, but under the "regular system of practice." I would not be understood as saying that all old school practitioners are of this stamp. Fr from it. Perhaps a great part of them are men of good medical information, but there are some among them more illiterate than even the ignorant Thomsonian they despise, and should be classed with him. Quacks belong to no particular class of men—there are some among all classes. There may be some among Reformers; if so, let them be exposed: but as a class of physicians, the name cannot be attached to them. Let the reformed practice testify to the truth of this assertion. However, in many places, the name of an "Botanic physician," exposes him to the suspicion of quackery, and that of an "Old School physician," clears him from it, however deserving. The reformed practice has been traduced; but let us, in retaliation, give a candid view of that of the old school.
It may be worth observing that the "old school" of medicine critiqued by the Worthingtonians' journal in 1836 was the still-embryonic, chemistry-centric new school of Dr. Vage's 1799 discussion.
Update (June 15, 2022): A figurative instance of "Old School" from 1685
A search of Early English Books online turns up one interesting instane of "Old School" that seems to anticipate the later figurative use of the term to mean "traditional." From Robert Boyle, Of the Reconcileableness of Specifick Medicines to the Corpuscular Philosophy to Which Is Annexed a Discourse about the Advantages of the Use of Simple Medicines (1685):
And tho I might here entertain you with the Vertues of some other Simple Remedies, plentifully given, yet for brevity sake I shall rather observe in general, That I doubt not but several Simple Medicines (I speak of alterative not evacuating ones,) would be found far more effectual than they are commonly thought, if they were given in a much larger Dose, and continued for a competent time. And probably so many Physicians (especially of the Old School,) would not be so forward to reject either Specifick or simple Remedies, as having found some of them not to answer Expectation; if they would allow them as fair a Tryal, as they give to their own Prescriptions, such as the Chalybeats of the Shops, the Spaw, or Tunbridge Waters, the Decoctions of Guajacum, &c. which they often give with divers intermediate helps for a Month or six Weeks, and sometimes for two Months together, without expecting that in a few weeks, much less in a very few days, they should perform the cure.
Here, the "Old School" seems to be not a single physical school, but an older (and arguably outmoded), tradition-based school of thought or practice.