Yes. It is recommended to fast in the first day of Muharram as the narration from Imam Al-Redha (AS) asking the person who met him on the first of Muharram; are you fasting? Fasting on everyday in Muharram is good except on the tenth day of Muharram which was taken by Bani Umayyad as an Eid and they fabricated narrations on fasting that day of Ashura.According to Ahlul Bayt (AS) fasting in Ashura is Makrouh and it is called as the fasting of the enemies of Ahlul Bayt.
'Wassalam.