How much would rent cost each week for a small two bedroom apartment in New York?

It depends on the borough and on the neighborhood. In Manhattan in a nice safe neighborhood, a two bedroom apartment starts around $2500/month and goes up from there. In one of the outer boroughs, it will be slightly less. My daughter lives in Astoria, Queens and she pays $1350 for a one bedroom.

You mean Manhattan? 1600$, at the very least.

generally 1500-2000.

it can be quit prisey :/