The first part
of the hadeeth i.e. “There are no days more loved to Allah for you to worship Him therein than the ten days of Dhul Hijjah.” is Saheeh and has been reported through various ways in authentic narrations.
Regarding the second part
of the hadeeth “Fasting any day during it is equivalent to fasting one year and to offer salatul tahajjud (late-night prayer) during one of its nights is like performing the late night prayer on the night of power. [i.e., Laylatul Qadr].”
Then this part is Da’eef as per al-Hafidh Ibn Hajar, mentioned in “Fath al-Baari” (which is an explanation of Saheeh al-Bukharee), Volume 2, page 461.
Imam Tirmidhi classed it as “Ghareeb” since it is not narrated except from Masood ibn Waasil from an-Nahaas, who is unknown in the sight of al-Bukharee, and as-Saadir al-Manawee and others consider him to be weak.
Ibn al-Jawzee (rahimahullaah) said that the hadeeth is not true and is narrated only from Masood ibn Waasil from an-Nahaas and Masood is Da’eef (weak) and he has been weakened by Abu Dawud.
Regarding an-Nahaas, al-Qataan said that he is rejected and Ibn ‘Adee said that he is not worthy at all.
[Source: "Takhreej ahadeeth 'Ihya Uloom ad-Deen”, volume 2, pages 617-618]
Al-Baghawee (rahimahullaah) declared its chain to be weak. [Source: Sharh as-Sunnah 2/624]
Shaykh al-Islam Ibn Taymiyyah (rahimahullaah) classified it to be da’eef (weak) [Source: Sharh al-'Umdah 2/555]
Shaykh al-Albani (rahimahullaah) has classified the above hadeeth to be da’eef as well. He classified it as da’eef in his checking of Ibn Majah (1728) , Da’eef at-tirmidhi (123), Saheeh wa Da’eef Sunan at-Tirmidhi (758), Da’eef al-Jaami’ as-Sagheer (5161), Da’eef at-Targheeb wa Tarheeb (734).
He also mentioned the second part of the hadeeth as “Da’eef in its totality” in his book, “Silsilah Ahadeeth ad-Da’eefah” Volume 11, page 242 no. 5142.
And Allaah Knows Best.