Does a Faraday cage block magnetic field?

No. The point of a Faraday cage is that it's made of a conductor, which responds to electric fields. A strong magnetostatic field is different, and will barely be affected by the Faraday cage. (The cage may have some magnetic properties, but that's not what makes it a Faraday cage, and it's unlikely to have a significant impact.)

There's a little info on Wikipedia about magnetic shielding: http://en.wikipedia.org/wiki/Electromagnetic_shielding#Magnetic_shielding

If you want to block a magnetic field, a faraday cage made of mesh is a bad choice.

You would need a cage made of solid metal sheets. A thick enough sheet would completely block the field on the other side.

This site has a calculator that draws the magnetic field across a metal sheet: https://www.kjmagnetics.com/thickness.calculator.asp

(it is from a company that sells magnets).

You will need to make a screen of high permeability magnetic material, which is not itself permanently magnetised. An alloy with the commercial name Mu-metal has been used for this purpose.