What did Al-Sha'rawi mean by his statement: "The day does not precede the night except in one act of worship, which is standing at 'Arafah, for the night that comes after it is an annex to the Day of 'Arafah"?

What the Sheikh means is that the night precedes the day. So, the night of Friday comes before the day of Friday, and similarly for all other days. If the crescent moon is sighted at the end of Sha'ban after sunset, then that night is the first night of Ramadan, and the Tarawih prayers are performed, followed by the first day of Ramadan. If the crescent moon is sighted at the end of Ramadan after sunset, then that night is the night of Eid, and no Tarawih prayers are performed on it. The night of Nahr (sacrifice) is an exception; it is the night of Eid, but it follows the Day of Arafah, which ended with sunset. Therefore, it is permissible to perform the standing (at Arafah) during this night before the dawn breaks.