No

Ruling from mtg Gamepedia:

If a spell moves itself into another zone as part of its resolution (as Arc Blade, All Suns' Dawn, and Beacon of Unrest do), rebound won't get a chance to apply

Moreover, the card exiling itself is part of the resolution, so Cast Through Time will not even be able to affect it as it would be already exiled

No

Ruling from mtg Gamepedia:

If a spell moves itself into another zone as part of its resolution (as Arc Blade, All Suns' Dawn, and Beacon of Unrest do), rebound won't get a chance to apply

Moreover, the card exiling itself is part of the resolution, so Cast Through Time will no longer be giving it Rebound when Rebound would trigger, as the card is already exiled.

Ruling from mtg Gamepedia:

If a spell moves itself into another zone as part of its resolution (as Arc Blade, All Suns' Dawn, and Beacon of Unrest do), rebound won't get a chance to apply

Moreover, the card exiling itself is part of the resolution, so Cast Through Time will not even be able to affect it as it would be already exiled

No

Ruling from mtg Gamepedia:

If a spell moves itself into another zone as part of its resolution (as Arc Blade, All Suns' Dawn, and Beacon of Unrest do), rebound won't get a chance to apply

Moreover, the card exiling itself is part of the resolution, so Cast Through Time will not even be able to affect it as it would be already exiled