Of American-English origin, the phrase to go for the jugular, and its variants, mean: to criticise or attack somebody aggressively or decisively; to target an adversary’s weakest or most vulnerable point.

The image is of attacking a person fatally in the throat or neck, where the jugular vein runs.

The phrase to go for the jugular occurs, for example, in the account of a match between the Australian tennis player Nick Kyrgios (born 1995) and the Russian tennis player Daniil Medvedev (born 1996), by Matt Majendie, published in the Evening Standard (London, England) of Monday 5^th September 2022:

WIMBLEDON runner-up Nick Kyrgios produced a vintage display to knock out defending US Open champion Daniil Medvedev in a high-quality fourth-round match in New York overnight.
The match-up promised to be the standout tie of the tournament so far—and it did not disappoint, as Kyrgios, enjoying the best tennis of his career, won 7-6, 3-6, 6-3, 6-2.
Spectators at Flushing Meadows will be hard-pressed to find a better set of tennis than the opener, which the Australian edged 13-11 in the tiebreak.
Despite dropping the next set, when the Australian fell into his usual trap of allowing the demons to creep in, he then went for the jugular against his Russian opponent, whose excellent defensive game was just no match for the aggressive, mercurial Kyrgios.

The earliest occurrences of the phrase to go for the jugular and variants that I have found are as follows, in chronological order:

1-: From the Atchison Daily Patriot (Atchison, Kansas, USA) of Friday 17^th January 1879:

THE SENATORIAL SKIRMISH.
The Senatorial Outlook in Kansas Sadly Beclouded by the Swarm of Candidates.
Anderson Looming Up as the Coming Man.
The Way a Newspaper Correspondent Views the Situation.

Special to the Kansas City Journal.
Topeka, Jan. 16.—The members are strangely non-committal on the senatorial question, evidently being in doubt as to where lightning will strike, and are wanting to be on the winning side, of course. In the absence of any positive knowledge in the matter, the relative strength of the leading candidates may be safely set down as follows, at this date, although political situations change as rapidly as the wind sometimes:
[…]
Horton, and Anderson, and Sid Clarke, and even Simpson, have votes lodged in Anthony as a convenient deposit until such time as Ingalls’ back is broken, and then, with the characteristic sincerity of politicians, they will go for the jugular of Geo. T., provided he does not in the nick of time utilize their confidences to his own profit. It would be a clear case of embezzlement wouldn’t it?

2-: From the account of a speech about the corruption of the police deparment of New York City, delivered on Friday 7^th September 1894 by the U.S. clergyman and social reformer Charles Henry Parkhurst (1842-1933), and published in the Democrat and Chronicle (Rochester, New York, USA) of Saturday 8^th September 1894:

“So Devery is dismissed. That is good. It is also good that the other two have been dismissed and all such things are very satisfactory. But the dismissal of two or three captains does not go to the root of the matter. Their dismissal goes to prove the entire rottenness of the whole department, but it does not eradicate or cure that rottenness. When the whole body is discussed, through and through, it does not affect a cure to cut off an arm here and a leg there. No, that won’t do. In cases like that, you must go for the jugular. In this case we must go for the jugular.”

3-: From one of the unconnected paragraphs making up the first and second columns of the fourth page of the Owensboro Daily Messenger (Owensboro, Kentucky, USA) of Wednesday 15^th September 1897:

At only two appointments in his swing around the circle did Col. Hindman have audiences to greet him. That was at Benton, where Charlie Wheeler met him in joint debate, and at Russellville, where John Rhea went for his jugular. If the colonel only wants to be heard for his cause it might be well to take a free silver speaker along.

4-: From the account of a speech that W. J. Crawford, of the Farmers’ Union, delivered at the court house, published in The Kingfisher Times (Kingfisher, Oklahoma, USA) of Thursday 12^th May 1910:

He gave a graphic description, commencing below the grass roots, of the manner in which the Corporation Law was conceived, begotten, born and brought into existence, written and implanted into the constitution. He then described how the predatory interests made a tiger leap at the jugular vein of the corporation law.

5-: From the portrait of one Col. M. G. Campbell, in a letter by a certain N. C. West, published in The Kosciusko Herald (Kosciusko, Mississippi, USA) of Friday 26^th May 1911:

In all his writings and speeches he goes for the jugular vein of his adversaries and holds his auditors with salient and thought-laden expressions.

6-: From “Skin ’Em Alive” Jim Reed Is After the Hide of Hoover, about James Alexander Reed (1861-1944), United States Senator from Missouri, published in The Brooklyn Daily Eagle (Brooklyn, New York, USA) of Sunday 6^th January 1918:

Reed is regarded by many judges as an able Senator and a man whose influence in the Senate might be really very large if he could overcome his natural propensity to leap for the jugular.

7-: From The Same Tammany ¹, published in the New York Tribune (New York City, New York, USA) of Saturday 11^th June 1921:

The persistent source of the city’s misgovernment is Tammany. So it is now and so it has been for nearly a century. All paths of corruption in this city lead to Tammany Hall. […]
[…]
Beginning in 1897 the city’s public began to realize that Tammany must be ousted as the first step to better government. Slowly but steadily the public eye became fastened on this main thing. It seemed as if a struggle of a century had been won. But three years ago its foot slipped and Tammany came back. Inevitably corruption came back with it, and it has grown worse as Tammany has become bolder.
So the Meyer committee ², no matter what its original desire, must leap for Tammany’s jugular.

¹ Tammany was the name of the central organisation of the Democratic party in the City (formerly also in the State) of New York, located in Tammany Hall. The name Tammany came to be especially associated with the corruption which characterised at various times the government of New York.
² Meyer Committee was the colloquial name of the Joint Legislative Committee to Investigate the Affairs of the City of New York, whose mandate was to inquire into the administration of the various departments of the city government, and to propose remedial legislation.