Duskmantle Guildmage

Whenever a card is put into an opponent's graveyard from anywhere this turn, that player loses 1 life.

Does the first ability wear off in the cleanup step like a continuous effect?

Or does it actually last until the actual turn end because of the delayed trigger?

It doesn't end at the start of the cleanup step, because it's not a continuous effect.

As you say, it is a delayed triggered ability - it will trigger when its triggering event happens. “This turn” here doesn't have the special meaning it has for continuous effects; it simply adds a check that is done to know if the triggering event happens or not.

A situation where it's relevant: after Duskmantle Guildmage's ability resolves, Giant Growth is cast on an Eager Cadet controlled by the opponent, then Night of Souls' Betrayal is cast. At end of turn, Giant Growth's effect will end, then Eager Cadet dies, which triggers Duskmantle Guildmage's delayed triggered ability.

Daniel Kitachewsky
L3, Paris, France
Rules Netrep