"ANW" is timed, so being fast obviously helps. Speed isn't just important for your finish time, though; it can also prevent counterproductive dithering. There's a place for in-game analysis, but it's a slippery slope toward overthinking. In general, a ninja's movements should be fluid.
Running is important for speed training, and it's also a good all-purpose cardio exercise. Still, "ANW" rarely involves straightforward sprints. Ninjas need to be fast at everything, from scaling walls to navigating a rope maze. Try picking up the pace at various points in your workout, but be careful: Not only does more speed give you less time to react, but rapid, jerky movements are more likely to injure you than make you a ninja.