Unlike the original game, where double Aku Aku protection was automatically granted at the start of the boss fight, this is not the case here: you must pass the fight without ever getting hit.
You will find yourself on a stone slab that forms a kind of chessboard. Be careful never to fall into the water all around. Ripper Roo will hop around this board, making TNT box icons appear at regular intervals. After a while, he will blow them up. To avoid this attack, simply stay on the part of the board that Ripper Roo doesn't touch. Then Ripper Roo will start hopping again, but this time it will bring up Nitro box icons. Avoid walking on them, or they will explode.
Again, stay on the part of the board that Ripper Roo doesn't touch and when he blows up the boxes, he will be stunned for a few moments. This is the opportunity to attack him, quickly use one of your attacks to steal a notch of life energy from him. To defeat Ripper Roo, you have to repeat the process two more times. The third time however, the portion of the board not affected by the explosive boxes will be very small. You will have to position yourself carefully and maybe jump during the explosion, in order not to get hit. Hit Ripper Roo three times to defeat him and go up to the next Warp Room.