In this topsy-turvy political landscape, it’s perhaps unsurprising that a pair of Ivy Leaguers are fighting it out for a Texas Senate seat, but it certainly is surprising that the liberal Democrat is being given a fighting chance.

Rep. Beto O’Rourke, the fresh-faced, onetime punk rocker and Columbia grad, is crisscrossing the Lone Star State selling a message that hasn’t resonated there for years: a future with free health care, more immigrants and fewer guns. The congressman from El Paso spreads his platform in a pickup truck and has plenty of cash.

Indeed, his fundraising is a match for incumbent Republican Sen. Ted Cruz, a former Princeton debating champ and presidential candidate who acknowledges he’s in his toughest political battle.

Mr. Cruz is starting to pop up in Texas towns as often as Mr. O’Rourke, looking to head off the Democrat’s surge.

While the Real Clear Politics polling average gives Mr. Cruz a 4-percentage-point advantage, it has moved the race into the toss-up category. Mr. Cruz no longer enjoys the double-digit lead he held in some polls when summer began. The tight race is even drawing President Trump, who says he will hold a rally for Mr. Cruz in October.

Mr. O’Rourke has benefited from an almost adulatory press, and there’s no question he is tapping into widespread Democratic animus toward Mr. Trump. While he doesn’t use the word “impeachment,” the notion that the president is guilty of “high crimes and misdemeanors” is something Mr. O’Rourke makes clear he takes as proven.

But his campaign has not been free of hiccups.

The Cruz camp has gleefully plastered Mr. O’Rourke’s mug shot in social media posts, an unflattering portrait taken by Texas authorities after his driving while intoxicated arrest in 1998. Mr. O’Rourke, then 26, was traveling at “a high rate of speed” in a 75 mph zone when he hit another car and, according to a witness, tried to flee the scene.

That was one of two brushes with the law for the younger Mr. O’Rourke, who was also arrested in 1995 on an attempted burglary charge. Neither arrest resulted in a felony conviction, and the DWI was vacated after Mr. O’Rourke completed a court-ordered program.

Still, a young man with beer and a pickup truck is hardly a disqualifying image in Texas, and to anyone reading the coverage of Mr. O’Rourke’s bid, it would seem an enhancement. In state and local accounts, he is invariably portrayed as a charismatic, 45-year-old idealist who galvanizes enthusiastic crowds.

Mr. Cruz, on the other hand, is presented as a more dour figure, although at 47 he is essentially the same age as Mr. O’Rourke.

“I think the dynamic in Texas is that you have an energetic new face against a person in Cruz whom Texans know well and conservatives support, but that warmth is not there,” said Cal Jillson, a professor of political science at Southern Methodist University.

Both men serve in Congress, but Mr. Cruz has the higher profile that comes with the Senate — and he is using that to his advantage this week with his defense of Mr. Trump’s Supreme Court nominee, Judge Brett M. Kavanaugh.

Mr. Cruz picked apart Democratic objections, arguing that they were rooted in a desire to fuel their party’s “most angry activists.”

In a radio appearance Tuesday, Mr. Cruz did not cite Mr. O’Rourke by name, but he drew clear lines between his position on Judge Kavanaugh and that of liberal Democrats, who “all want to vote ‘no.’”

Doing so, Mr. Cruz said, would put them “wildly out of step with the state they represent,” alluding to polls that show clear majorities in most states favoring Judge Kavanaugh’s confirmation.

“I’m going to vote ‘yes’ because the people of Texas want a strong, principled constitutionalist on the court who will defend our fundamental liberties,” Mr. Cruz said. “Who will defend free speech, who will defend religious liberty — and who will defend the Second Amendment.”

Mr. Cruz is hitting traditional conservative favorites, running ads urging Texans to stand when the national anthem is played and pointing to Mr. O’Rourke’s defense of pro football players who choose to kneel.

Conservative groups are rushing to his defense, with the Club for Growth vowing to spend liberally to back Mr. Cruz.

Mr. O’Rourke, meanwhile, has energized liberal activists who have gone years without having reason to be enthusiastic about their chances in a Texas senatorial race.

The last Democrat to win a Senate race in Texas was Lloyd Bentsen, in 1988. The last Democratic victory in any statewide race in the state was in 1994.

Boosting Mr. O’Rourke is an unorthodox yet folksy approach that appears to be tapping into voters well beyond the liberal base.

He swears profusely, so much so that the Cruz campaign used it in an ad and The Texas Tribune declared “f-bombs are part of his brand.”

In a widely watched YouTube video, Mr. O’Rourke speaks at a backyard barbecue in March.

“We’re putting party over country, ideology over results, presidential ambition over the needs of the people that we should be serving and representing,” Mr. O’Rourke says. “And I was stuck in this limo in D.C. wondering, what the f– are these guys doing in Washington, D.C.?”

The crowd greets his remarks with hearty chuckles.

Last month, after a San Antonio supporter sent him a skateboard, Mr. O’Rourke tweeted a photo of himself smiling as he clutched it.

“You need a little, like, moment,” he tweeted. “So this is going to be our Zen. Right here. This skateboard.”

That evening, the campaign put out a video that went viral of Mr. O’Rourke skateboarding in a Whataburger parking lot. His hometown paper, the El Paso Times, quoted approving responses from what it called supporters, with one saying it “conveys an athletic free spirit and a youthful energy.”

Mr. O’Rourke’s biggest surprise has been his adept fundraising.

He had almost $14 million on hand at the end of June, according to campaign finance records. That left him with more money than the Cruz campaign, which reported $9.3 million in the till on the same date.

The O’Rourke campaign boasts of an army of small givers, suggesting a true grassroots network of financial support. One liberal activist raised enough money on GoFundMe to place a mobile billboard displaying a nasty Trump tweet about Mr. Cruz — or “Lyin’ Ted” — outside the president’s rally for the senator.

Yet Mr. O’Rourke has received nearly 60 percent of his money from large, individual contributions, according to the Center for Responsive Politics. Mr. Cruz has taken in one-third of his money from similar contributions.

Mr. Cruz’s struggles to put distance in the race compare unfavorably with Gov. Greg Abbott, a Republican seeking re-election, who holds a near 20-point lead in most polls.

But Mr. Abbott, with $40 million in his campaign account, could help Mr. Cruz. Mr. Jillson said that consequently, “the messaging down the stretch is going to be overwhelmingly Republican.”

The reason Mr. Cruz does not enjoy the same polling margins as Mr. Abbott is simple, according to Mr. Jillson: “He’s lovable and Cruz is not.”

But Texas is still Texas, Mr. Jillison said, and Mr. Cruz’s reputation as a fighter and his conservative bona fides will prove decisive.

“I don’t expect [Mr. O’Rourke] to win at the end of the day, and I suspect the polls right now show him as a little closer than he really is,” Mr. Jillson said.

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