You can use a gate (Sonitus) to cut the snare off abruptly:
Set the Attack to 0.
Set Hold to the desired length of the snare (50 to 300mS).
Set Release to zero or near zero for the abrupt cut-off.
Set the Lookahead to 5 to 10 so you don't loose the attack of the snare.
Put a fast release compressor after the gate. The compressor will make the volume remain constant during the decay of the snare sound.
The sound you are referring to in the song is a sample, most likely two samples.
One for the "boom" sound (snare top) and one for the "shhh" sound (snare bottom).
This is what you would do for a real snare drum.
Use one mic for the top and one mic for the bottom. Reverse polarity on bottom mic.
Gate the top mic and use the gated signal as a side chain trigger for a gate on the bottom mic.
The side chain keeps both gates in sync for a tighter snare sound.
Compress both mics individually for desired sound.