I don't believe fat lot of good was ever in common usage with a positive meaning.
"A fat lot of good"
The earliest reference I can find in Google Books is 1876's Investing Uncle Ben's Legacy: A Tale of Mining and Matrimonial Speculations by Old Boomerang aka John Richard Houlding:
Non-ironic "fat lot"
As FumbleFingers comments, it's worth noting that a fat lot was also used non-ironically quite a bit earlier, for example in 1833: "adds a fat lot of knowledge" (or could this be ironic?). It's notably used when talking of livestock 1834: "The best fat lot of stots was purchased by Mr. Norrie"; 1862: "A fat lot of 100-lb. Sheep"; 1862: "A good fat lot of sheep".
But Bradford Frazee's 1845 An improved grammar of the English language lists these as incorrect constructions of adjectives:
He made a good pair of shoes. She knit a fine pair of socks. I gave a cold cup of water. James wears a new pair of shoes. He built a high string of fence. Susan bought a splendid pair of gloves. They bought a good load of hay. He drove a fat lot of hogs. They drive a fine span of horses. They have a most elegant supply of furniture. He wears a very fine suit of clothes.
Ironic "fat lot"
However, the use of fat lot (without of good) used ironically in slang also has heritage.
1844's The Downside Magazine and Monthly Miscellany says:
FAT. This adjective denotes privation or negation, and is so far equivalent to the alpha privative of the Greeks, but they differ inasmuch as fat includes an idea of contempt. Thus when a person speaks very knowingly upon any subject, he is silenced by being politely informed " that he knows a fat lot about it;" and sometimes, in a game at hand-ball, when a player entertains the idea that the plea of "hinderance" has been unfairly alleged by the other side, he expresses his opinion that it was a fat hinderance—an assertion indeed, which, when made with reference to certain stout young gentlemen, students in the college, would doubtless be true in the ordinary sense of the word "fat."
John Camden Hotten's 1865 A dictionary of modern slang, cant, and vulgar words : used at the present day in the streets of London, the Universities of Oxford and Cambridge, the Houses of Parliament, the Dens of St. Giles, and the Palaces of St. James : preceded by a history of cant and vulgar language from the time of Henry VIII, shewing its connection with the Gipsey tongue : with glossaries of two secret languages, spoken by the wandering tribes of London, the costermongers, and the patterers says:
FAT, rich, abundant, &c.; "a Fat lot;" "to cut it Fat," to exaggerate, to show off in an extensive or grand manner, to assume undue importance; cut up Fat," see under Cut. As a Theatrical term, a part with plenty of Fat in it, is one which affords the actor an opportunity of effective display.
An example from 1866:
... and I tink, Mr. Gatty, if I vork him properly, I can persuade him to advance our screws.'
'A fat lot of advance you'll get with a nipper like Naylor in the shop!' quoth surly Mr. Gatty.
And from 1869:
"We couldn't do, sir, if it wasn't for what we get from the parish ---"
" And that's a fat lot !" scornfully snorted the other old woman.
Ironic "much" and back to Shakespeare
The 1875 edition of William Gifford's The works of Ben Jonson; with Notes Critical and Explanatory, and a Biographical Memoir is illuminating.
The Reverend Peter Whalley had edited The Works of Ben Jonson in 1756 with footnotes, which Gifford added to.
On page 110-111 (or 408 in this viewer), Jonson wrote in Every Man In His Humour:
Brai. Ay, sir, there you shall have him. [Exit Know.] Yes — invisible!1 Much wench, or much son!
The footnotes add:
1 Yes — invisible! Much wench, or much son !] [...] What follows is proverbial; Much was a term of various senses, and often used as an expression of disdain and contempt. Much good may they do you, both wench, and son, if you find them. WHAL.
I know not what to say of Whalley's note. [...]
Much! is an ironical exclamation for little or none, in which sense it frequently occurs in our old dramatists. Thus in Heywood's Edward IV.
" Much duchess ! and much queen, I trow ! "
And in Shakspeare,
Is it not past two o'clock ? and here's much Orlando !
How do we link this lot to fat lot? Well, on page 185 we have a third note!
P. 110. Much wench, or much son.] There is a vulgar exclamation, '* A fat lot !" which represents precisely the same idea.