The phrase the university of life denotes the experience of life regarded as a means of instruction, in contrast to formal (higher) education.
—Synonym: the school of (the) hard knocks.

The phrase the university of life is now often used with the implication that life experience is of greater benefit than formal education. The following, for example, is from Death by Arrangement (London: Macmillan, 1972), by the British author Laurence Meynell (1899-1989):

The spires of Oxford could go on dreaming, the dons of Oxford go on cracking the laboured jokes in their little ivory towers till kingdom come for all he cared; he set about getting himself a degree in the university of life.

The earliest occurrences of  the phrase the university of life that I have found are as follows:

1-: From the address that Cornelius Conway Felton (1807-1862), Professor of Greek Literature, delivered during the meeting of the Association of the Alumni of Harvard College, held at Harvard College, Cambridge, Massachusetts, on Thursday 20^th July 1854—as transcribed in the Boston Semi-Weekly Advertiser (Boston, Massachusetts, USA) of Saturday 22^nd July 1854:

Another associate, of former days, I see no more:—the late Professor Greenleaf, ours by adoption—who, after a youth of toil and patient suffering, gave the rich maturity of his years, to the elucidation of the Law, in this place; the priceless value of whose instructions is now acknowledged, as far as the widening boundaries of our republic extend: who, not born to affluence, and not bred up to scholarly studies, achieved an honorable scholarship, in the university of life.

2-: From Notices to Correspondents, published in The New York Ledger (New York City, New York, USA) of Saturday 7^th November 1857:

Public School Boy.—Education is an infinite subject, and that is the reason why no definite answer can be given to questions respecting best modes of conducting it. But language is the fountain of it; and therefore we begin to learn language or languages for a foundation […]. When this knowledge is acquired in youth, a good foundation is laid, and this is all that schools and universities do, or, indeed, ought to do. After this elementary instruction comes the practical school, or university of life, in which the superstructure of a man’s destiny is reared.

3-: From the transcript of the lecture that the Rev. Mr. Alger, of Boston, Massachusetts, delivered during the annual meeting of the Massachusetts Teachers’ Association, held at Fall River, Massachusetts, on Monday 23^rd and Tuesday 24^th November 1857—transcript published in the Daily Evening Traveller (Boston, Massachusetts, USA) of Wednesday 25^th November 1857:
—In this text, however, the university of life denotes the highest level that can be reached in the school of life:

The world […] is a vast school house, well provided with teachers, and having mankind for pupils, and we are sent here to equip ourselves for the great work we have to perform to lead us to the knowledge of the Divine truth, and to fit us for a higher and brighter existence beyond this life. In considering this school of life, the duties and occupations are important. The literature of the world is ours for inspection, and the book of nature affords additional means for a thorough education. The mind that can appreciate such teaching, will find “books in the running brooks, sermons in stones, and good in everything.” Everything in psychology exists in the soul, and all neurology pervades the material world around us. Logic, the logic of life, belongs as much to John Smith as to Scotus Erigena.
The stream cannot rise higher than the fountain—so the spirit and desires of men can never rise higher than the fountain of God’s goodness. […] What are the true aims of the pupils in this great school of life? Oh, how few attain these, and how many means for securing them are neglected! Some of these aims ought to be the development of our natural powers—the right use and exercise of our spiritual faculties. The mass of mankind graduate in a primary school; a goodly class go into the grammar department; a select few reached the high school, while but a portion of these enter the college, and a yet smaller representation become residents in the university of life. When the work of education is done, when the man becomes an inheritor of heaven, his pupilage is consummated. And this is the great aim and object of all education in the world.

4-: From the review of Oxford University Reform, by Goldwin Smith, published in Oxford Essays (London: John W. Parker and Son, 1858)—review published in The Dublin University Magazine: A Literary and Political Journal (Dublin, County Dublin, Ireland) of August 1858:

Has the great intellectual reform of 1807 worked practically well? We attempt to answer the question by certain intellectual statistics, carefully gathered from the class lists. Let us look at the various classes from 1807 to 1847—forty years.
In those forty years, there have been 457 firsts, a fraction over eleven per annum. How many of these eleven have been in any degree conspicuous in afterlife? [list of names]
Total, First Classes, 457
           Notables, 80
This makes it evident that an Oxford first class is absolutely a test of power. Given a first-class man, with prolonged life and health, and the odds are almost equal that he will shine among his fellows in some way: be it as clergyman, lawyer, statesman, or man of letters.
But test these classes relatively. In the same forty years there were, as nearly as possible, 920 second classes, or exactly 23 per annum. Now, these eminent second-class men, whom we have marked, are [list of names]:
Total, Second Classes, 920
           Notables, 42
The numerous third class—probably some 1,400—do not give more than fifteen or twenty notables [list of names]. The fourth class, perhaps 1,600 strong, may yield some sixteen notables, belonging, in nearly every case, to what is known to the initiated as the honorary fourth. [list of names]
It would appear, then, that on the whole, the university of life confirms the decisions of the schools; that the proportion of distinguished men in the several classes is not far off the proportion who are crowned with more serious laurels. While, to encourage those who fail, it would appear that a “second” […] occasionally takes a “first” in life; and that “outsiders” in the third and fourth sometimes leave firsts “nowhere!” But surely these statistics are worthy of note, and, on the whole, endorse the University system as a test of general power and promise.

5 & 6-: From transcripts of the speech that the Rev. George T. Day, of Providence, Rhode Island, delivered during the anniversary of the Literary Societies of Maine State Seminary, held at Lewiston, Maine, on Wednesday 22^nd July 1863:

5-: From the transcript published in the Lewiston Daily Evening Journal (Lewiston, Maine, USA) of Thursday 23^rd July 1863:

The subject was in substance the relations of the discipline of letters to the discipline of life, showing the advantages of each kind of training, and the necessary connection of the one with the other. No man is really and truly educated till he has exercised his powers. The arguments urged in favor of each kind of training, were succinctly and pithily stated, and both pleas admitted to be forcible and unanswerable, but neither necessarily conflicting with the other. The school of letters gives breadth of intellect, the school of life gives intensity. The former trains; the latter makes practical. The former secures accuracy, the latter promptness. The former is the polished shaft, the latter the drawn bow and quiver. […]
[…] The orator concluded with a beautiful tribute to the lamented Mitchell, a graduate both of the school of letters and the University of life, the scholar and the soldier.

6-: From the transcript published in The Freewill Baptist Quarterly (Dover, New Hampshire, USA) of October 1863:

The silent teachers in the library stand motionless on the shelves, but when we touch them they distil their lessons into the soul with marvellous effect; and the vocal teacher in the lecture-room keeps us hanging upon his lips as he will, while his thoughts flow along the channels of our being till the very banks are overflown; but when we pass into the living world of men, where antagonists contend, and rivals struggle, and athletæ run; where merchants manage, and politicians plot, and generals manoeuvre; where enterprise strikes into the forest, and treachery lies in wait for its prey, and piety sends up its patient prayer to heaven, straight through all the jargon of men;—then we find ourselves in another school, where the teachers fill all the air with shouting, where reveries are broken by the push of violence, and lessons come to us in the form of deeds. One is the school of letters; the other is the university of life.

7-: From the review of Fifty Years’ Biographical Reminiscences (London: Hurst and Blackett, 1863), by the British Army officer and author Lord William Pitt Lennox (1799-1881)—review published in The Sporting Magazine (London, England) of August 1863:

It is the pleasantest book for a sportsman, for a country-gentleman’s library, or for a graduate anxious to obtain a double-first in the great university of life, that has been published for many a-day. It conveys pleasant information about everybody of any claim to notoriety that has appeared in London society for the last half-century, jockeys and jailors, actors and tailors, peers and pedagogues, millionaires and adventurers, princes and pugilists, duchesses and demireps, and is an encyclopædia of everything fashionable, from Almack’s to Aunt Sally, from the opera to the strolling players, from a banquet at the Carlton to a whitebait at Blackwall, from a Cowes regatta to a pull on the river, and from a masquerade to a dancing tea.

8-: From Chronicles of the Schönberg-Cotta Family (London: T. Nelson and Sons, 1864), by the British author Elizabeth Charles (née Rundle – 1828-1896):

Many of us had completed our academical course, and were already entering the larger world beyond—the university of life.