The word "than" is not a preposition. It is a conjunction after which a verb and sometimes a subject are commonly omitted, but understood, because they are the same verb or subject as in the preceding sentence.

Therefore all you really need to do is understand that the portion of the sentence following "than" is itself a sentence. Just supply the missing words, and the correct case will shine forth.

"I run faster than he."

is really:

"I run faster than he runs."

and not:

"I run faster than him runs."

Therefore "he" is correct.