All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is well-known for the scholars of Ahlus-Sunnah that Al-Husayn ibn 'Ali was killed on the day of 'Aashooraa', the 10th of Muharram. This is what many scholars have stated, like Al-Waaqidi, Abu Na'eem, Al-Fadhl ibn Dakeen, Ibn Abu Shaybah, Khaleefah ibn Khayyaat, Al-Bukhari (in At-Taareekh Al-Kabeer), Ibn 'Asaakir (in Taareekh Dimashq), Ibn 'Abdul-Barr and Al-Khateeb Al-Baghdaadi .
This is also the authoritative opinion of many late scholars like Ath-Thahabi, Ibn Katheer, Ibn Hajar and others . Some of them stated that Al-Husain ibn 'Ali was killed in the month of Safar; this is what Al-Waaqidi reported from Muhammad ibn Ka'b Al-Quradhi both, and then refuted this saying, “It is confirmed that he was killed on the day of 'Aashooraa' (the 10th of Muharram).” Al-Qurtubi reported that there is a consensus on this and said, “The scholars agreed in a consensus that he was killed on the day of 'Aashooraa' (the 10th of Muharram) in the year 61 A.H.”
They stated that he travelled from Makkah to Iraq on the 8th of Muharram. Based on this, he travelled the distance in one month and two days and not in less than a month as stated by the questioner; this is a sufficient period of time for travelling this distance.
For more details on the view of Ahlus-Sunnah on the tragedy of Karbalaa', please refer to Fatwas 87406 and 84152.
Allaah Knows best.