Just buy a ton of uzi ammo, and do a drive by; the car has a relatively low health pool. Drive up parallel to him, as he leaves the casino, and fire until the health bar depletes. If you destroy the car, and he escapes, run him over. The goal is to kill him as soon as possible, thereby ending the mission.
Another thing you can do is unlock Ray Machowski, complete "Arms Shortage" and get access to the M16, which destroys cars in seconds. Then go back to Asuka, trigger the mission, drive up to the blue marker at the casino, armed with the M16. Then, as he drives away, get out of your car, and shoot at him. Should go down in seconds, but act quick; he drives fast.
Note for second method: you can access the M16 for a very short period of time at the end of "Under Surveillance", after you kill the last round of guys in the balcony; their weapons should drop, but only for a short period of time. Obtaining them is possible if you're quick. You might have to sacrifice some health to get to them in time before they disappear.
Hope it helps.