I would say both sentences are technically correct but the first sentence sounds a bit more natural :)

Though it might sound weird, I will say the second one is correct. My sister has shorter hair than I (have shorter hair). Since you are comparing two subjects — my sister and I — you should use the subject pronoun “I” instead of the object pronoun “me”. If you are comparing object with object, you should then use the object pronouns. For example, “Tom loves Jerry more than me” is very different from “Tom loves Jerry more than I”. The first sentence is saying that “Tom loves Jerry more than he loves me”, whereas the second sentence is saying that “Tom loves Jerry more than I love Jerry”.