Or he's stripped of citizenship and deported by Trump

https://www.theguardian.com/us-news/2025/jul/01/trump-zohran-mamdani-citizenship

I think the state legislature and governor will be more of a hurdle to clear than council. Since in US and Canada, most municipal powers are rooted in state/provincial legislation, they have the ability to override or veto, sanction or outlaw anything the city tries to implement. Gov. Hochul is an establishment Democrat so I expect some efforts to water down the most radical policies, but aside from her characteristic flip-flopping she won't try to put herself in the way of city politics.

I think it's less about what he's capable of doing and more about what he signifies.
Electing an openly socialist mayor to one of the biggest cities in the country is a huge step, one that could gain momentum and they definitely do not want more of that happening, especially in the house and Senate.
They think if they nip it in the bud now they may be able to get ahead of it, but I think they fail to realize they might make a political martyr out of him instead.

Hopefully, he will be able to overthrow the city council so he can enact his policies.

I'm no expert in New York City governance; I'm not even on the same coast as New York. West Coast, Best Coast.

With that said, NYC's size and structure is not too dissimilar to that of a US State, save for a unicameral legislative body (New York City Council). Matching that, the Mayor of NYC is the head of the executive, with powers to appoint commissioners to various agencies and civil/criminal courts, as well as executive functions like administering city services like fire departments, police, and tax collection.

Meanwhile, the 51-member Council is headed by the Speaker, who presides over the body and controls the order that legislation is considered. So far as I can tell, the members are elected by district, every four years, so that each district has roughly the same population. So far, these procedures parallel those of US State governments.

As for the interplay between the Mayor and the Council, the defining criteria of any government is how it achieves its policy objectives, in passing the budget. Like with the California Governor, the Mayor's office will propose -- and later execute once duly-passed -- the budget and the Council will consider and approve or reject it. The final budget is sent to the Mayor for ratification, but can also be vetoed. In this case, the Council can vote to override a mayoral veto.

So for the titular question, with regards to only the structure of the government of NYC, yes, the Council could very much block much of what a future Mayor Mamdani wants to achieve. The Council could do this by passing laws that mandate minimum fares for transit, forcing tax breaks for the wealthy, and anything else that directly counters his policies. But he could veto such laws, and the Council would have to muster some 2/3 of the votes to push it through.

In turn, though, a future Mayor Mamdani could potentially use his executive control to direct the transit system to vary (read: change) the tariff structure so that bus routes in less well-off neighborhoods become free. Within the parameters of existing law, the Mayor could also instruct the Police Chief to do (or not do) certain things, and this wouldn't be within the Council's direct control except that they could have a Council committee do an investigation and raise new legislation. But that goes back to what the Council can and can't do.

Essentially, there's a fair amount of ground for a progressive NYC Mayor to deliver campaign promises, except that the budget and existing laws will require working with the Council. But as a practical matter, if a future mayor wins a substantial fraction of the city-wide vote, it would be strange that 2/3 of the Council could be in staunch opposition.

And that budget vulnerability can actually be a negotiating tactic. Here in California, setting aside any broader opinions about the policies and wisdom of the currently second-term Governor of California, he managed to negotiate a bill to cut red-tape for housing (or roll-back environmental laws, depending on who you ask) and tie it to the state budget, due end of June. So when push comes to shove, when the budget is coming due, there would suddenly be room to negotiate, even with bitter enemies. No one respects a government that cannot pass a budget on-time.

I personally am of the opinion that when a legislative body wishes to obstruct, or when an executive wants to pursue a policy, then neither should half-arse it. A future Mayor Mamdani should force the Council to publicly reject what he wants to put forward, each and every time. Let the people of NYC see who is actually fighting for the citizenry, and who is kowtowing to monied interests. Commentators often talk about "spending political capital" when doggedly pursuing a policy, but that's kinda the job: do it right, or step aside and let someone else do it. NYC deserves the best mayor they can get.

New York state has a whole bunch of power over the city too. I think the city had real bad finances a long time ago and part of bailout by the state can't with a whole bunch of power.

Mayor of New York isn't the end goal