Praise be to Allah, and may peace and blessings be upon the Messenger of Allah.
It is authentically established from the Prophet ﷺ that the Hour will not begin except upon the worst of creation, and that Allah تعالى will take the souls of the people of faith before it begins through a pleasant wind.
The Prophet ﷺ said:
“The Hour will not begin until it is no longer said on earth: Allah, Allah.” Narrated by Muslim.
In the hadith of Abdullah ibn Amr رضي الله عنهما, it is mentioned that Allah تعالى will send a cold wind from the direction of Syria, and it will take the soul of everyone who has even the weight of an atom of faith in his heart. After that, only the worst of people will remain. Then Satan will command them to worship idols, and after that the Trumpet will be blown, causing the people to swoon.
Accordingly, the Hour will not begin upon the righteous believers. Rather, it will begin upon the worst of creation after the souls of the people of faith have been taken.
As for the words of Allah تعالى:
﴿The greatest terror will not grieve them, and the angels will receive them, saying: “This is your Day which you were promised.”﴾ [Al-Anbiya: 103],
and His words:
﴿And on the Day when the Trumpet will be blown, those in the heavens and those on earth will be terrified, except those whom Allah wills.﴾ [An-Naml: 87],
these verses do not mean that the righteous believers will witness the Blast that causes the people to swoon at the end of the world, because the texts indicate that the souls of the people of faith will be taken before it occurs.
The scholars have mentioned that the greatest terror referred to in the verse may occur at the Blast of Resurrection, or may refer to the terrors of the Day of Judgment and the stages of reckoning. It is not necessarily the terror that occurs at the end of the world.
Allah تعالى says:
﴿And the Trumpet will be blown, and whoever is in the heavens and whoever is on earth will swoon, except those whom Allah wills. Then it will be blown again, and behold, they will be standing, looking on.﴾ [Az-Zumar: 68].
Thus, the blowing of the Trumpet occurs in two stages: the Blast that causes all living beings to swoon at the end of the world, followed by the Blast of Resurrection, after which people will rise for the reckoning.
In conclusion: The believers will not witness the actual beginning of the Hour; rather, Allah will take their souls before it, and then the Hour will begin upon the worst of creation. As for the believers being protected from the greatest terror, this concerns the terrors of the Day of Judgment and what follows the Resurrection, not the end of the world and the Blast that causes all living beings to swoon.
And Allah تعالى knows best.