The British colloquial phrase to come the acid—also, in later use, to come the old acid—means:
– to behave in an unpleasant, aggressive or overbearing manner;
– to speak in a sarcastic or caustic way.

The image is of someone behaving, or speaking, in an acid manner—i.e., in a cutting, sharp or hurtful manner.

I think that the adjective old was added on the pattern of the earlier phrase to come the old soldier over someone, meaning to use one’s greater age or experience to deceive someone or to shirk a duty.

The phrase to come the acid originated in British Army slang during the First World War. Its two earliest recorded occurrences are from:

1: “A Funny Story.”, by R. B. James, published in The Londoner. The Journal of the 1/25th Battalion The London Regiment (Vol. II. No. 1. May to August 1917):

Bill was always knocking ole Pip about,—keeping him in order, he called it. Not really knocking him about, of course, ’cos he could’ve broken ole Pip’s back with his thumb and finger, in spite of Pip’s being two inches taller. Six one he was and about as thick as a bootlace, the sort you buy in the bazar—and about as strong. You’d see him lying on his bed in the bungalow, with a blanket over him,—it was never very hot in Bangapet—and Bill come in singing and shouting, and however sound asleep ole Pip seemed to be he was out of bed on the other side like a flash before Bill reached him. Bill said it was bad for him, lying about all the afternoon, and he wasn’t going to have it. Led poor ole Pip a nell of a life he did, but my word, let anyone else try to come the acid on Pip and wasn’t there a schemozzle [note 1], not half!
—source: 25th County of London Cyclist Battalion The London Regiment, by Simon Parker-Galbreath

2: Tommy’s Dictionary of the Trenches, appended to “Over the top” by an American soldier who went Arthur Guy Empey, machine gunner, serving in France (G. P. Putnam’s Sons – New York and London, 1917), by the American author, actor, screenwriter and film producer Arthur Guy Empey (1883-1963):

“Coming the acid.” Boasting; lying about something.

The phrase appeared in The Daily Mail (Hull, Yorkshire, England) of Saturday 31^st December 1921, which reproduced What the Soldier said, a record of war slang by Edgar Preston, published in The National Review (London, England) of December 1921:

The “Old Man” was the Colonel, and “coming the old man” was said of anyone inclined to be fussy and to make too much of a little brief authority; “coming the acid” meant boasting, bragging.

Edward Fraser and John Gibbons recorded the phrase in Soldier and Sailor Words and Phrases (George Routledge and Sons, Ltd. – London, 1925):

Acid, coming the : Stretching the truth : making oneself unpleasant : trying to pass on a duty : exaggerating one’s authority.

The earliest instances of to come the old acid that I have found are from a sarcastic book review published in The Sporting Times (London, England) of Saturday 22^nd October 1921—the phrase occurs in the book that was reviewed, What Woman Wishes (Hutchinson & Co. – London, 1921), a novel by the British philosopher, sociologist and social critic Anthony Mario Ludovici (1882-1971):

THE VISCOUNT AND THE NYMPH.
WHEN MONICA BIT JIMPER IN THE LEG.
By ARAMIS.

Natural curiosity about the ornamental sex impelled me to read “What Woman Wishes,” by Anthony M. Ludovici. This author is not a professional humorist, but he gets there unconsciously. Mr. Ludovici, I gather, has the soul of a haughty aristocrat, still lamenting the death of Charles I., but he shows most cordial good feeling towards plebeians. In fact, the heroine of his latest book, Miss June Penrose (hereinafter called simply Jimper), has not an “h” to her name, and indulges in language which is distinctly low. Her manners and customs are the more clearly outlined because of her association with the English nobility. Charles Highbarn, Viscount Chiddingly, the Commander Kenworthy [note 2] of a new Tory party, is “all over” Jimper, and his demned [sic] distinguished old Father, Lord Firle, treats her as one of the family.
Jimper is a Notting Dale girl who knows wot’s wot:—
“Had she not been courted by almost every well-set-up young man in the Dale, and had she not resisted each in turn? It was known that ‘nothing in trousers’ dared ‘to come the old acid’ with her, and men would have as soon attempted to assault a steam-engine as place an arm round her comely waist. She was now twenty-three, and no man—aye, no man who ever breathed, except her father and her uncle Solo—had ever been known to kiss her.”
                                                        The Amorous Amazon.
I refuse to credit this. Jimper had been engaged to a piou-piou [note 3] and had spent some time with his family in France. However, when another French artillery man arrives in Notting Dale he does actually “come the old acid” with her (I hope it is nicer than it sounds). Father and Uncle Perkins find the hefty wench in a fainting condition, and Mrs. Perkins accuses the gallant Marcel of “playing dirty French tricks.” Jimper arises from her swoon in indignation. “‘’Ave yer gone skatty?’ she says. ‘What’s the matter with yer? ’E’s all right. ’E only kissed me. Can’t I come over dithery if I want to?’” The shocked relatives concede to the amorous Amazon the right to dither.

 

Notes:

 

1: Here, the noun schemozzle denotes a quarrel, a row. This word is first recorded as shlemozzle in the portrait of a Jew of London, in Houndsditch Day by Day (Sands & Company – London, 1899), by the British journalist and author Arthur Morris Binstead (1861-1914). The word may be from Yiddish shlimazl, meaning misfortune, unlucky person (cf. the noun schlimazel, denoting a consistently unlucky or accident-prone person), with subsequent reduction of schle- to sche-.

 

2: This possibly refers to Joseph Kenworthy (1886-1953), Lieutenant-Commander in the Royal Navy.

 

3: In France, the noun piou-piou designates a private soldier. The primary meaning of piou-piou, a French word of imitative origin, is the cry of young chicks.