How can one atone for a sin after causing the death of a cat due to frightening it, bearing in mind the story of the woman who entered Hellfire because of a cat, and also repenting for raising dogs?

Inciting animals against each other is impermissible. What the accused did might fall under this category, and he must repent and feel remorse for his actions, for "remorse is repentance," and "one who repents from sin is like one who has no sin." There is no specific expiation for this act in Islamic law; what is obligatory is repentance, and it is recommended to do good deeds or give charity.