In the first phase of this fight Red will fly around tossing bombs at you once more (similar to the pre-orange fight) which are easy enough to avoid if you keep moving. on occasion he will slow down and move to the bottom corner of the screen, when he does this he is about to dash across the bottom of the screen, simply jump over him though the timing can be a little tricky the first time.
In the next Phase of the fight Red will walk towards you, jumping and using an extended kick to try and hit you. This is easy enough to run/slide under. After this he will begin to shoot you using two methods, one of which is his hardest to avoid and most dangerous attack. The first type of shot has two fireballs criss-crossing as they move across the screen, it should be reasonably easy for you to place yourself as to avoid them. the single shot however will curve up if you try and jump over it too early, try and leave it to the last possible moment before jumping over this. Often you'll get hit but thankfully the rest of the fight/red's attacks are easy enough so taking a little damage here shouldn't be a big issue.