The effect doesn’t wear off until just before your next untap step (even if an effect will cause that untap step to be skipped).

The player is chosen as the ability resolves. Once a player is chosen, it’s too late for other players to respond by casting spells.

The targeted opponent may choose himself or herself.

The targeted opponent may choose you. In that case, only you can cast spells and attack with creatures before your next turn begins.

Players other than you and the chosen player may still perform game actions besides casting spells and attacking with creatures: They may block with creatures, activate abilities, perform special actions, and so on.