That's a rather odd setup - normally a "universal motor" is used in this situation (like a DC motor but instead of permanent magnets the stator has field windings that connect in parallel to the brushes so that the magnetic fields reverses as the armature current reverses, so that the motor rotates the same way regardless of the polarity of the supply (no need for a bridge rectifier).
You could control the DC output of the bridge rectifier to control this motor, but remember the voltages involved are dangerous (they can and do kill, especially DC). Using high-voltage MOSFETs (about 300V rating perhaps) to control this would involve careful design and protection components so that if the MOSFET dies it doesn't destroy the Arduino or electrocute anyone.