How much does it cost to move a 1 bedroom apt in NYC?

It depends very much on what is in the apartment, and who you hire to do it.

If you have a lot of nice, expensive, fragile things, then you should probably hire a professional mover - obviously this is rather expensive in New York.

If you don't have any antiques or chandeliers though, and you feel like a couple of strong college students could get the job done, you might want to consider a service like Agent Anything.

On agentanything.com, you can post a Mission, like "Help me move my 1 bedroom apt," and specify a price that you would be willing to pay some college students to do it, or accept offers from students.