Can Mel Soul Reflection legend use her activated ability on your opponent's turn

No. Soul's Reflection's activated ability — "Disempower me, [T]: Give a unit at a battlefield −2 [S] this turn" — has no Reaction tag, so it follows the standard rule that activated abilities can only be used on your own turn in an Open State (Rule 381; FAQ #8332).

For contrast, her other ability ("When you empower something else, empower me") is a triggered ability and does fire on your opponent's turn — but the activated ability itself cannot be used then, not even during a Showdown.