Grading: Its chain is severely weak.
Technical note / Defect ('illah): The reasons for its weakness are the weakness of 'Amir bin Salih bin Rustam, and that Musa, the father of Ayyub, is "majhul" (unknown). Furthermore, the narration of Ayyub's grandfather 'Amr bin Sa'id bin al-'As (who is known as "al-Ashdaq") from the Prophet (peace and blessings be upon him) is "mursal", because according to Imam al-Bukhari (al-Tarikh al-Kabir 1/ 422) and Imam al-Tirmidhi his hearing from the Prophet (peace and blessings be upon him) is not established; and he is not the 'Amr bin Sa'id who was a Companion.
Note / Clarification: On the basis of these two defects, Imam al-Dhahabi called it "mursal da'if" in "al-Talkhis" and declared 'Amir bin Salih al-Khazzaz "wahin" (very weak).

Reference / Source: It was narrated by Imam Ahmad (24/ 15403/ 1), al-Tirmidhi (1952) and 'Abd Allah bin Ahmad in "Zawa'id al-Musnad" (15403/ 2, 16710, 16717) through various routes from 'Amir bin Salih.
Grading: Imam al-Tirmidhi says that it is "gharib" and we know it only from the narration of 'Amir bin Abi 'Amir al-Khazzaz ('Amir bin Salih bin Rustam), and in my view it is "mursal".

Reference / Source: Imam al-Tabarani in "al-Kabir" (13234) and "al-Awsat" (3658), and Ibn 'Adi in "al-Kamil" (6/ 211), transmitted this narration as marfu' through the route of Muhammad bin Musa al-Sa'di, from 'Amr bin Dinar (the steward of the family of al-Zubayr), from Salim, from Ibn 'Umar (may Allah be pleased with him): "No father has given his child an inheritance better than good manners."
Technical note / Defect ('illah): Ibn 'Adi declared it defective because of Muhammad al-Sa'di and called him "munkar al-hadith". Moreover, there is agreement on the weakness of his shaykh 'Amr (the steward of the family of al-Zubayr).

Corroborating chains and supporting reports: Imam al-'Uqayli in "al-Du'afa'" (1772) transmitted a similar narration from Abu Hurayrah (may Allah be pleased with him) through the route of Mahdi bin Hilal, from Hisham bin Hassan, from Ibn Sirin.
Technical note / Defect ('illah): Mahdi bin Hilal was called a "kadhdhab" (liar) by several imams, so he cannot be relied upon. Al-'Uqayli says that it is not "mahfuz" (preserved) from the narration of Hisham bin Hassan; rather it is known only from the narration of 'Amir bin Abi 'Amir al-Khazzaz, and this hadith is not established.