The most ideal way to frame this sentence would be: I can run faster than he (can). However, 'than he (can)' and 'than him' are both defensible. 'Than him' is less formal and professional. Ideally, 'he' is used if a verb follows it.
Look at these examples: I am taller than he is, I eat more than he does, I had more chickens than he had, I want this more than he wants etc. But, 'than he' without a verb following does not appear to be heard very often.
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