Is the sacrifice sufficient for someone residing in the Sultanate of Oman if he shares in it with his family in Egypt, and it is slaughtered on the day of Eid al-Adha in Egypt, which coincides with the Day of Arafah in Oman?

The validity of an Udhiyah (sacrificial animal) depends on the location of the slaughter, not the location of the one offering the sacrifice. Therefore, if the Udhiyah is slaughtered after the Eid prayer in the country where it is being sacrificed, it is valid on behalf of the one offering it, even if the time for slaughter has not yet begun in their own country.