Does coil require extra resistor before connecting mains?
I suspect this scheme will not work because 50Hz/60Hz is too fast. You will just get a vibrating rod with very little travel. A better scheme might be to use a speaker and audio amplifier to drive it at a lower frequency. Remove the speaker cone, or direct connect to the driver coil where it connects to the cone. It is worth doing this because it is simple, safe and you can test whether the scheme is viable. What volume of air do you want to pump?