Can I use Sett Brawler in Showdowns?

No, you cannot activate Sett, Brawler's ability during a Showdown.

According to RiftJudge FAQs (specifically FAQ #8543 and #3969), Sett, Brawler's ability does not have the ACTION or REACTION keywords. Because of this, it is considered "standard speed" and is subject to the following strict timing restrictions:

• It must be your turn.
• There must be no chain present.
• You must be outside of a Showdown.

Because a Showdown is an active game state where these conditions are not met, you cannot use his ability to spend a buff while a Showdown is occurring. You must wait until you are outside of a Showdown and it is your turn to activate the ability.