First, that link says 80 kg. That requires about 800 N to lift on earth.
There is no spec on that page that indicates the current required at 12 V. You need to know this to design the circuit. If you can't dig around to find the spec, then you will need to measure a sample. DC current is strictly due to the voltage applied to the DC resistance of the coil. You can use a ohmmeter, or otherwise measure the current and voltage at some operating point. The current will scale linearly with voltage, so you can compute what it will be at 12 V.
Once you know the current, you pick a switching transistor that can handle it and the voltage. A low side N-channel MOSFET is the obvious choice. At such a low voltage, they are available with quite low on resistances. This is important since the FET on resistance (Rdson) not only causes a voltage drop, but will cause power dissipation in the FET proportional to the square of the current. The easiest would be a FET that has low enough Rdson that the dissipation is low enough at your current to not require a heat sink.
For example, let's say the magnet draws 5 A at 12 V (it's DC resistance is 2.4 Ω). A 20 mΩ FET would dissipate (5 A)2(20 mΩ) = 500 mW. A TO-220 case, for example, would get hot in free air, but should be able to handle it. Check the FET datasheet carefully to see what the dissipation will be, the temperature rise due to that dissipation, and whether that resulting temperature is acceptable or not.
PWM is good for modulating the coil current. However, make sure the flyback catch diode is rated for the current. At this low voltage, use a Schottky for the lower forward drop and therefore less wasted power.
Make sure the PWM period is short enough so that the current changes only a little during one period. Think of the rippling current as the average DC plus a AC component. Only the DC component produces the magnetic field you want. The AC component only causes additional I2R heating and thereby wasted power.
Fortunately a magnet like this has substantial inductance, so even modest PWM frequencies will have sufficiently short periods. For example, let's say the magnet coil has 10 mH inductance (just picked something out of the air, may not match your magnet). (12 V)(40 µs)/(10 mH) = 48 mA. That is how much the current would change thru the coil when 12 V is applied to it for 40 µs. That's small compared to the 5 A we used as example above. Again, you have to determine the real numbers and do your own calculations. The worst case is a square wave with the on time being half the PWM period. For a 40 µs PWM period, the on time would be 20 µs, and the ripple 24 mA peak to peak.
40 µs for the PWM period is the highest I'd use. That's 25 kHz and just above the audible range, but still "slow" for any competent microcontroller and FET driver switching time. You should be able to do 50-100 kHz without any real drawback.