There is a hadith with very similar wording in Sunan Ibn Majah, the Book of Zuhd, out of which a quote is provided below:
عَنِ ابْنِ مَعْقِلٍ قَالَ: دَخَلْتُ مَعَ أَبِي عَلَى عَبْدِ اللَّهِ فَسَمِعْتُهُ يَقُولُ قَالَ رَسُولُ اللَّهِ ﷺ: النَّدَمُ تَوْبَةٌ
It was narrated that Ibn Ma'qil said: I entered with my father upon 'Abdullah, and I heard him say: The Messenger of Allah (ﷺ) said: "Regret is repentance."
— Sunan Ibn Majah, Book 37, Hadith 4393
There are other hadiths that convey the same meaning with different wording, e.g., in Sunan an-Nasa'i 4068 (grade: Sahih) and Sunan Abi Dawud 4273 (grade: Sahih).
As for the specific hadith in Sunan Ibn Majah, it has been documented in multiple books (e.g., Musannaf Ibn Abi Shaiba Hadith 179, Musnad Ahmad Hadith 4113, Sahih Ibn Hibban Hadith 613, Al-Mustadrak Hadith 7695, among others). The hadith is authentic (grade: Sahih) through multiple narration chains (hence being in Al-Mustadrak), as in Musannaf Ibn Abi Shaiba ('Abdullah Ibn Mohammad → Waki' Ibn Al-Jarrah → Sufian Ibn Sa'eed → 'Abdul-Karim Ibn Malik → Ziad Ibn Abu Maryam → 'Abdullah Ibn Ma'qil and his father Ma'qil Ibn Muqrin → 'Abdullah ibn Ma'ood) and in Musnad Ahmad (Ahmad Ibn Mohammad → Kathir Ibn Hisham → Furat Ibn Sulaiman → 'Abdul-Karim Ibn Malik → Ziad Ibn Al-Jarrah → 'Abdullah Ibn Ma'qil and his father Ma'qil Ibn Muqrin → 'Abdullah ibn Mas'ood).
The narration specific to the hadith in Sunan Ibn Majah, though, has Hisham Ibn 'Ammar in its chain — who is trustworthy, but in his old age his memory became weak, so this hadith narration chain is graded Hassan.