Roughly a year ago, Zohran Mamdani sipped a cup of tea at a Yemeni cafe in Queens and spoke of what seemed like an unimaginable ambition: He wanted to become mayor of New York City.

Months later, he declared his candidacy, beginning the race with nothing more than optimism. He had few endorsements and name recognition that he put at “1 percent of New Yorkers knowing who I am, and that’s a charitable estimation,” he said in an interview on Wednesday.

In the ensuing months, Mr. Mamdani managed to create what turned out to be an indomitable campaign for mayor, exciting young voters with a brand of creativity, social media wizardry and a never-wavering message about the necessity of making New York more affordable.

He emerged from Tuesday’s Democratic primary for mayor as the likely winner, building a coalition of support that crossed racial, economic and religious lines across most of the city.

On Wednesday, Mr. Mamdani took a rare moment to rest from his breakneck campaign. After “about four hours of sleep,” he was home in his Assembly district in Queens taking congratulatory calls from policymakers like Representative Hakeem Jeffries of New York, the House Democratic leader, and reflecting on his success and the challenges that still lie ahead in the general election in November and beyond.

His apparent victory, he said, was a “reflection of a hunger across New York City, across neighborhoods, both ones that are described as progressive and ones that are not, for a different kind of politics, for a new generation of leadership, and for a democracy where New Yorkers can see themselves and see their concerns and see their struggles reflected back.”