No, a standard Nintendo DS USB cable will not work with an Action Replay DS.
Different Functions: The standard DS USB cable is designed solely for transferring save files and game data between the DS and a PC. It lacks the hardware interface to communicate with the Action Replay cartridge.
Proprietary Connection: Action Replay DS uses a specific cartridge-slot connection to load its software and cheat databases. This requires a dedicated link.
Warning: Using third-party cheat devices can void your Nintendo warranty and carries a risk of save corruption. Proceed at your own risk.