Praise be to Allah.
The hadith “Whoever recites Soorat al-Ikhlaas one thousand times on the day of ‘Ashoora’…” is not a saheeh hadith, and it has no basis. It is a report that is often quoted in the books of the Shi‘ah, from their book al-Iqbaal bi’l-A‘maal, by ‘Ali ibn Moosa ibn Taawoos. The author of this book narrated it without any isnaad, from Ja‘far as-Saadiq.
Shaykh al-Albaani (may Allah have mercy on him) said:
I do not know of any proven hadith which speaks of the virtue of reciting “Qul Huwa Allahu ahad” one thousand times. Rather everything that has been narrated concerning that is very flimsy (waahin jiddan)." (Silsilat al-Ahaadeeth ad-Da‘eefah 6/333).
The virtuous deed on the day of ‘Ashoora’ that is proven in sound reports is fasting.
It was narrated that Ibn ‘Abbaas (may Allah be pleased with him) said: I never saw the Prophet (blessings and peace of Allah be upon him) making sure to fast on any day, regarding it to be superior to other days, except this day, the day of ‘Ashoora’, and this month – meaning the month of Ramadan. Narrated by al-Bukhaari (2006) and Muslim (1132).
Ibn al-Qayyim (may Allah have mercy on him) said:
None of the hadiths about wearing kohl, wearing fine clothing, spending generously (on the family), and praying and doing other virtuous deeds on the day of ‘Ashoora’, are saheeh, not even one hadith. Nothing is proven from the Messenger of Allah (blessings and peace of Allah be upon him) concerning that day except the hadith which speak of his fasting. Anything other than that is false." (Al-Manaar al-Muneef p. 103).
And Allah knows best.