Can Sett spend his buff during Showdown?

No, you cannot spend Sett, Brawler's buff during a showdown.

According to RiftJudge FAQ #8763 and RiftJudge FAQ #9127, Sett, Brawler's ability ("Spend my buff: Give me +4 might this turn") is a base-speed ability. Because it lacks the [ACTION] or [REACTION] keywords, it is subject to strict timing restrictions:

• It must be your turn.
• There must be no chain present (the state must be open).
• You must be outside of a showdown.

Since a showdown is a combat state, you cannot meet these requirements while in one. You must wait until you are outside of the showdown and it is your turn to spend the buff.