His supporters show up at news conferences. Basic municipal services have gotten an unexpected dose of excitement. Celebrities help him promote his agenda. In the process, he has notched several significant early wins. He has also achieved a kind of truce, at least for now, with former President Donald J. Trump, a mercurial leader with a penchant for celebrities. But as Mr. Mamdani, a Democrat, marks an early milestone of his tenure, it remains an open question whether he can translate his star power into enacting the progressive policy proposals that got him elected. While he has fierce critics, many of whom still view his past criticism of the Police Department and Israel as deeply problematic, the mayor has managed to ease the concerns of at least some skeptics. “It’s early, but so far, so good,” said Jay Jacobs, the state Democratic Party chairman, who drew attention for not endorsing Mr. Mamdani during the campaign. “We may not agree philosophically on everything, but he’s doing the job.” As he approached Day 100, long established as a benchmark for assessing an administration’s early vision, his team set out to highlight a commitment to the day-to-day duties of the mayor’s office. While many of those tasks are typical of local government — trash pickup, snow removal and filling potholes — the 34-year-old mayor has leaned into his knack for creating viral content to raise interest and awareness about city programs. As a powerful storm approached the city this winter, Mr. Mamdani’s social media calls for additional snow shovelers helped recruit thousands of new workers. In another social media video, this time about the city’s emergency alert system, his office said more than 50,000 new users signed up for the program in just a week. Together with the W.N.B.A. star Natasha Cloud of the New York Liberty, Mr. Mamdani unveiled a bracket-style competition in which residents can vote for minor repairs they want the mayor to personally make in their neighborhoods. In just a few weeks, more than 21,000 votes poured in. And to raise interest in his 2-year-old child care program, Mr. Mamdani enlisted Cardi B to help judge a contest for a jingle that would become the program’s official theme song. “The challenge we set for ourselves was to work as hard and as fast as New Yorkers do,” Mr. Mamdani told reporters on his 99th day in office this week. His star status, however, sometimes draws backlash. During a bitter cold snap this winter, his surprise appearance on “The Tonight Show Starring Jimmy Fallon” struck some as tone-deaf, at a time when deaths among homeless people in New York were surging. “Too much style and posing,” said Curtis Sliwa, the Republican who ran against Mr. Mamdani in last year’s election, citing longstanding problems with street homelessness, public housing and infrastructure. Still, Mr. Sliwa, who attacked Mr. Mamdani relentlessly during the campaign but recently appeared with him in a comedy sketch at the annual Inner Circle show for City Hall reporters, gave him some credit, with caveats. “We had Eric Adams, a man with an attitude who would party until dawn, and now we have someone who seems to keep regular hours,” Mr. Sliwa said, referring to the previous mayor. “So having Zohran as an alternative, I think for many people, even if they disagree with him, it brings a certain stability.” Mr. Mamdani has also directed his huge audience to another routine, but crucial, reality of governing the city: the budget. Earlier this year, Mr. Mamdani held an unusually somber news conference at City Hall about a huge gap in the city’s budget, warning that either the state would have to raise taxes on the wealthy or he would be forced to raise local property taxes to balance the budget. The mayor’s public appearance was widely seen as an attempt to pressure Gov. Kathy Hochul of New York to approve a millionaires’ tax, a key priority for Mr. Mamdani and his base. Ms. Hochul, a moderate Democrat who is up for re-election this year, strongly opposes such a measure. After the City Council released its budget proposal, which suggested different ways to close the gap, Mr. Mamdani sharply criticized the plan and released a video calling out the Council speaker, Julie Menin, leading some of his supporters to attack Ms. Menin online. The city’s budget woes, still unresolved, could become a major obstacle for Mr. Mamdani in advancing his agenda. Andrew Rein, the president of the fiscally conservative Citizens Budget Commission, said Mr. Mamdani faced a “particularly difficult” budget process, but that his communication skills could help him navigate those waters. “We’ve seen him bring new strategies and tools to the way he communicates, but to a very practical, old-fashioned problem,” Mr. Rein said. “When he uses his communication skills to engage people more, improve government functioning and help New Yorkers understand what trade-offs need to be made, that will be a big win.” On the night of Mr. Mamdani’s election victory, hundreds of people filled the streets, some arriving spontaneously, waiting to see the mayor-elect leave his campaign headquarters. Campaign staff members leaving the building were greeted with cheers, by name, long after midnight. One attendee compared the street scene to Beatlemania. “I feel like I’m at a presidential inauguration,” said Medhavie Agnihotri, a 25-year-old tech consultant. “For the first time in a long time, I feel this kind of hope.” His star aura since then, at least in some circles, shows no signs of fading. Outside City Hall, New Yorkers and tourists often stop him for selfies, peering through the iron fence in search of the mayor. This week, on Day 97 of his tenure, a huge crowd gathered in the lobby of the bustling Bellevue Hospital in Manhattan, where Mr. Mamdani announced that the city would begin moving some people with serious medical conditions out of the notorious Rikers Island jail complex to the hospital. He entered the room to cheers and applause from a large crowd that appeared to include many hospital workers holding up cellphones to record the mayor’s remarks. Dozens more watched from elevated galleries. One of them, Ricardo Granados, a 67-year-old retiree, was on his way to take his son to a doctor’s appointment but stopped to see what the commotion was about. He lit up when he learned that Mr. Mamdani would appear, saying he had met him earlier when he was campaigning in his neighborhood. “I love him to death. I think he’s going to make a real difference,” Mr. Granados said. “He wants to find out who needs what and he wants to help.”