CR 611.2b tells us that continuous effects with “for as long as ~ remains tapped” (e. g.
Amber Prison,
Vedalken Shackles) won't do anything “if the ‘for as long as’ duration never starts, or it ends before the moment the effect would first be applied”.
What if I pay {4} and tap my
Amber Prison targeting creature A, then untap the artifact (e. g. with
Voltaic Key) before the ability resolves, and then pay {4} and tap
Amber Prison again targeting creature B? I'd say that both creatures don't untap during their controllers' untap step, because when either of the activated abilities resolves, it sees a tapped
Amber Prison, so the continuous effect is created and applies to creature A or B, respectively. The continuous effect shouldn't care about what happened before, it just should check at the very moment it is created, i. e. when the ability resolves.
Whether or not my guess is right, CR 611.2b should express this more accurately.
Mario Haßler