Last time I checked, if you rest/sleep you heal all your attributes (assuming that you have food and water). It doesn't matter what you did in the last quarter day.
So if you use the artifact in the 'Day' and at the end of that quarter day you get the attribute damage. Then you go to sleep (or rest) in the 'Evening' and will wake up with no damage.
Yes, that is sort of true. However, this doesn't take into account items that inflict damage once per QD. This conversation is based around that unknown element. So if you reason that healing damage occurs first, but the item's damage proc happens last, it will occur after you heal any other damage you might have (including damage you received last QD using the artifact) but NOT this recent damage proc.
Resting doesn't say that you have no damage after you rest, but that you heal damage you have. So if you spend a QD resting you heal whatever you had. But if the damage doesn't occur until the end of the QD, you didn't have the damage to be healed in the first place. It's the same as waking from a nap and immediately being punched by a companion.
If you want the damage from the artifacts to be meaningful and not something PCs can easily get around by taking a nap, this is one way to handle it. And the vagueries of the rules allow this to be a perfectly valid interpretation.
Some issues with this:
* This mean that you as the holder of the artifact know that you will be broken at the end of the quarter day (if that is the case).
* If more than one person is holding it at the end of the quarter day, who will get damage, the last one. Because if that is the case, then you can plan who will get damage after you have done the dungeon exploration (assuming here that such a task doesn't take more than six hours). Is time a factor (how much time he held the artifact during the exploring quarter day) or can you use toss it to someone that doesn't risk becoming broken just before the quarter day ends.
Maybe more, but I am tired at the moment.
1) I have no problem with the character breaking at the end of the QD. What is wrong with that? The lingering cost of the item finally takes its due...
2) Everyone who used the item gets the damage. How would it be any different from extracting the damage up front from multiple users? surely if you damaged them immediately upon use and they juggled the item you would damage them as they were using it, and not allow PC 2 to become immune to the damage just because PC 1 had already been damaged. Having the damage trigger later would be no different.