Yes. At least sometimes. Donald Trump became president of the USA, the first office he ever ran for. Silvio Berlusconi became Prime Minister of Italy and is still an important political player in the country until today. Manny Villar in the Philippines became Senator and Senate President. Petro Poroshenko in Ukraine, Andre Babis in the Czech Republic, Bidzina “Boris” Ivanishvili from Georgia and Thaksin Shinawatra in Thailand are further examples.
This being said, there are also billionaires who lose. Michael Bloomberg got himself elected mayor of New York, but his bid for the democratic nomination two years ago went nowhere. Not only did he lose (despite unlimited resources), but I doubt you would even remember an ad or tagline from his campaign. Long before him, Ross Perot from Texas and Steve Forbes also ran for president, and lost.
Money in politics is an intriguing topic. When I speak to politicians or give interviews to journalists, I noticed that they often want to hear an all or nothing answer. One the one hand, some want to hear that money is everything in a campaign and can create wonders at the polls. On the other hand, others like to hear the opposite, namely that money does not matter all that much in campaigns. the truth is somewhere in between: money is one of the handful of important factors in a campaign. It helps to win awareness, to amplify your message, and gives the freedom to continue a campaign endlessly. In itself, money does not communicate a message, however. It can not create enthusiasm and it does not replace a solid and winning campaign plan. It helps to execute it.