In order to have Saiki appear in Arcade mode you must have at least 40 target actions done before the encounter on stage 4. A simple way to do this is tone your difficulty settings down to very easy and do as many as you, lose rounds with characters if you need to. Your target goal per round (character) is at least 5, so that's about 15 a complete match.
Target Score for stage 4 : 2 500 000
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