Big difference, and it matters here.
In a sublet, you stay on the lease and the other person lives there. In an assignment or takeover, you're asking the landlord to release you and let the new person step into the lease. For a market-rate apartment, the landlord can generally charge a reasonable processing fee when they approve an assignment, but it should track actual out-of-pocket costs: running a credit check, reviewing the file, preparing a lease amendment, maybe using a third-party screening service. A few hundred dollars is on the steeper side, but not unusual. Ask for an itemized list.
The broker fee is shakier. If no licensed broker was involved and the leasing office is not performing brokerage services for the new tenant, calling it a broker fee is a reach. In New York, broker fees generally belong to licensed brokers for actual services. A managing agent can sometimes collect a fee if they have a licensed brokerage arm and they're actually acting as a broker, but just sending paperwork is not it. I'd ask them: what services are being provided for that fee, and who is the broker of record?
Caveat: rules vary with building type, rent regulation, and the original lease language. If you're rent-stabilized, the specifics can be different. If your lease already includes an assignment fee clause, that's their strongest argument. But one month's rent for zero work is worth pushing back on.