From The TimeSplitters Wiki

Requirements: Platinum: 1st with 5 Lives

Gold: 1st with 4 lives

Silver: 1st with 3 Lives

Bronze: 1st

In this arcade match its just you up against Mickey Two-Guns. The only two weapons avalible are the Garret Revolvers X2 and just the plain old Garret revolver. Mickey has high stamina and takes around five hits to take down while you take about two. When you begin the mission head over to the open area near the water fountain. Grab the Garret revolver thats located in front of the two small buildings. Then turn around and face the large building. Mickey will pop out either from the bottom windows or the door on the far left. While he's far away quickly shoot him as many times as you can. If you are lucky he will die but if not he will charge towards you. Go near the fountain and keep going moving around and circles. Mickey will keep shooting at you but if you keep moving he will keep missing. When he reloads his gun quickly shoot him down and grab is double revolvers. Killing Mickey should be easy now. If you get lucky Mickey may run away from you unarmed. You should quickly chase him and shoot him down before he gets to a weapon spawn.

Unlockables: Jared Slim and Mickey Two-Guns