I’m going to re-write this but keep it close to what your wrote, sort of clean it up a little. That doesn’t mean it will be the best version but hopefully it will be a little shorter and concise:
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When a cold and arrogant spy, — When an arrogant spy (No need for both cold and arrogant)

begins to experience memories of another life,–experiences memories of another life (no need to say begins)

he goes against orders — he disobeys orders (same thing, less space)

to search for the mysterious woman he?s sharing a past with, — To search for the mysterious woman he shares a past with (Try to avoid -ing verbs when possible)

but as he gets closer to tracking her down, — but as he gets closer (no need to say tracking her down, it’s implied) {Actually this whole line could be cut, I will try a version without it as well}

he discovers his own agency has made both of them a target in an attempt to stop them from meeting — He must evade his own agency who’s made him a target to keep them from meeting.
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“When an arrogant spy experiences memories of another life, he disobeys orders to search for the mysterious woman he shares a past with, but as he gets closer he must evade his own agency who’s made him a target to keep them from meeting.”
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When an arrogant spy experiences memories of another life, he disobeys orders to search for the mysterious woman he shares a past with, but must evade his own agency, who’s made him a target to keep them from meeting.”
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Hope that helped, good luck with this!