Mike Pence is in many ways Donald Trump’s opposite—an unfailingly polite and humble establishment Republican with a full head of naturally white hair and a wife of 31 years.

“I’m kind of the B-list Republican celebrity,” the vice-presidential nominee and Indiana governor told several hundred supporters in Leetonia, Ohio, last Wednesday. “I’m not the main event.”

He also has far better ties to mainstream Republicans, who now are counting on him to get the better of his Democratic counterpart, Sen. Tim Kaine of Virginia, on the debate stage Tuesday night. The stakes for that event have risen after Mr. Trump’s uneven debate performance last week against Democratic nominee Hillary Clinton, and after the publication of some of Mr. Trump’s tax records.

Much as Mr. Pence, 57 years old, is the stylistic counter to Mr. Trump, he also has taken the opposite approach in preparing for that showdown. While the New York businessman said he relied mostly on instinct going into his debate, Mr. Pence said he has spent several days in 90-minute mock debates, then studied video playbacks. A friend, Wisconsin Gov. Scott Walker, played the part of Mr. Kaine.

“My goal is to show that Donald, a doer in a game usually reserved for talkers, will bring the bold change sought by everyday Americans voiced in their aspirations and frustrations,” Mr. Pence said in an interview aboard his plane during a three-state swing recently. “And I’ll contrast what I did in Indiana to Sen. Kaine in Virginia.”