Play as Dixie, but do not lose Kiddy. Hit Arich two times by just holding the barrel, then wait for him to come down to get hit by it. Then, immediately switch places with Kiddy. Jump on Arich's back and get the barrel. You will not have to wait for or worry about him spitting balls at you. Next, run down almost as far as possible and throw the barrel upwards. Again, you will not have to wait for him to spit balls. Immediately put Dixie on your shoulders and throw her up to the branch. Get the barrel again and run almost as far as possible. Throw the barrel upwards and Arich will give up.