What is the authenticity of the حديث (Hadith) that mentions the demolition of the House (Kaaba) and its rebuilding better than it was, and that buildings in Mecca will rise above the tops of the mountains?

This narration attributed to Abdullah ibn Amr ibn al-Aas is not a Prophetic Hadith. It has been narrated through him via three routes:

The first route is from Ata al-Amiri with the wording: "How will you be when you demolish the House... then it will be rebuilt better than it was. So when you see Mecca with its springs gushing forth, and you see buildings towering over the mountain peaks, then know that the matter has overshadowed you." Its chain of narration is amenable to being considered hasan (good).

The second route is from Mujahid with the wording: "When you see buildings rise to Abu Qubays, and water flows in the valley; then take your precaution." Its chain of narration is weak, but Al-A'mash followed it up.

The third route is from Yusuf ibn Mahak with the wording: "When you see its houses – meaning Mecca – rise above its two prominent mountains, and its valleys gush with rivers, then the matter has drawn near." Its chain of narration is disputed.

The multiplicity of these routes indicates that the narration has an origin from Abdullah ibn Amr, but it is not to be taken as a marfu' (attributed to the Prophet, peace be upon him) narration due to the weakness of the chains of narration, Abdullah ibn Amr's renown for narrating from Isra'iliyyat (Jewish traditions), and the possibility of ijtihad (independent reasoning) in such reports. Thus, his saying might have been an ijtihad in understanding other hadiths, such as the hadith about "tall buildings."

It is worth noting that the demolition of the Kaaba and the flowing of water in the Arabian Peninsula are signs of the Hour that are established by authentic Prophetic hadiths.