Can I activate Sett Brawler to give him +4 for the cost of the buff when my opponent is playing Singularity in his turn, targeting my Sett Brawler?

No, you cannot activate Sett, Brawler's ability in response to your opponent playing Singularity.

According to RiftJudge FAQ #7445 (related to Sett, Brawler), Sett's ability ("Spend my buff: Give me +4 might this turn") does not have the [ACTION] or [REACTION] keyword. Therefore, it is a standard-speed ability.

Standard-speed abilities can only be activated during your own Main Phase when the state is open (i.e., when there is nothing on the chain). Because your opponent has played Singularity, it is currently on the chain, meaning the state is closed. You cannot activate standard-speed abilities while the chain is not empty.