Sure, a magnet bouncing through a coil will generate power. You can see plenty of LED flashlights that work like this.
An 1800mAh - I assume you didn't mean mega-amp-hour ;) - battery is pretty big though, and it would take many hours of shaking to charge one. Super-rough calculation: a 20g magnet moving at 2m/s has 40mJ of kinetic energy, and lets say you managed to recover 50% of that, for 20mJ of electricity per stroke. 1800mAh at 3.6V for a Li-Ion battery is 23.3kJ so you need 1.2 MILLION strokes to charge the battery. Say it is bi-directional and you can shake it at 5Hz, that's producing 20mJ pulses at 10Hz... so 32.4 hours of constant vigorous shaking to charge the battery.
The voltage you get will be a function of the magnetic field strength (how good is your magnet), the speed of the magnet through the coil and the number of windings on the coil. Refer to Maxwell's equations...
The current will be approximately the voltage produced by the coil, less the voltage of the battery you are trying to charge, divided by the total impedance of everything (coil, battery, etc) in that loop. It might also be limited by the inductance of your coil.
The current so produced will cause a dragging force on the magnet. If the current is too small, the magnet will whizz straight through the coil and produce no power, but if the current is too large, the magnet will stall in the coil. You will need to design the coil and charging circuit in relationship with the mass of the moving magnet so that it extracts a reasonable quantity of kinetic energy per motion.