Will the person who intends to fast 'Ashura during the day be rewarded for his fasting?
Question

I know of the virtue of fasting the day of 'Ashura, and that it is expiation for the year that came before. But because we use the Gregorian calendar I did not know when 'Ashura is until the morning of the day itself. I had not eaten anything, so I intended to fast. Is my fast valid, and will I get the reward of this day and receive expiation for the previous year?.
Praise be to Allah.for making you keen to do nafl acts of worship. We ask Him to make us and you steadfast in doing that. 
With regard to your question about forming the intention to fast from the night before, there is sound evidence from the Prophet (peace and blessings of Allah be upon him) that the intention to observe a nafl fast formed on the morning of the day itself is valid, so long as a person has not ingested anything that breaks the fast after dawn. ‘Aa’ishah (may Allah be pleased with her) narrated that the Prophet (peace and blessings of Allah be upon him) entered upon his family one day and said: “Do you have anything (i.e., any food)?” They said, “No.” He said, “Then I am fasting.” (Muslim, 1154, 170). The word “idhan” (translated here as “then”, in the sense of “in that case”) indicates that it is permissible to form the intention of observing a nafl fast on the day itself. This is unlike the obligatory fasts, which are not valid unless one forms the intention the night before, because of the hadeeth, “Whoever does not form the intention to fast before dawn, has not fasted.” Abu Dawood, 2454; al-Tirmidhi, 726; classed as saheeh by al-Albaani in Saheeh al-Jaami’, 6535. What is meant here is obligatory fasts. 
Based on this, then your fast is valid. With regard to getting the reward for fasting, will this be reward for the whole day or from the time of the intention only? Shaykh al-‘Uthaymeen (may Allah have mercy on him) said: 
There are two scholarly opinions on this matter. The first is that he will be rewarded from the beginning of the day, because the correct Islamic fast can only be from the start of the day. 
The second view is that he will only be rewarded from the time of his intention, and if he formed the intention at midday then he will be rewarded for half a day. This is the correct view, because the Prophet (peace and blessings of Allah be upon him) said, “Actions are but by intentions, and each person will have but that which he intended.” This man only formed the intention during the day, so he will be rewarded from the time he formed the intention. 
Based on the correct opinion, if a fast is connected to a particular day, such as fasting on Mondays and Thursdays, and fasting on al-ayaam al-beed (the 13th, 14th and 15th of each Hijri month) or fasting three days each month, and a person forms the intention during the day, he will not be rewarded for the entire day.  Al-Sharh al-Mumti’, 6/373. 
The same applies to the case of one who did not intend to fast 'Ashura until after dawn had come: he will not get the reward for fasting 'Ashura’, which is the expiation of one year, because his case is not like that of one fasted the whole day of 'Ashura’, rather he only fasted part of it, from the moment he first formed the intention of fasting. 
But he will be rewarded in the general sense of the reward for fasting during the month of Allah Muharram, which is the best kind of fasting after Ramadaan (as it says in Saheeh Muslim, 1163). 
Perhaps one of the main reasons for your and many people’s not knowing when 'Ashura is, or al-ayyaam al-beed, until the day has started, is – as you mention – use of the Gregorian calendar. Perhaps missing out on these special days will motivate you and others whom Allah blesses with guidance, to follow the lunar Hijri calendar which Allah has prescribed for His slaves and approved of for His religion, even if that is only with regard to their personal dealings, so as to revive this calendar which reminds us of different shar’i occasions, and in order to be different from the People of the Book whom we are commanded to differ from and be distinct with regard to their rituals and unique characteristics, especially since the lunar calendar was followed by the earlier Prophets, as may be understood from the hadeeth which mentions the reason why the Jews fasted 'Ashura – which is a day that is known through the lunar calendar. They said that it was the day on which Allah had saved Moosa. This indicates that they followed a lunar calendar and not the months of the solar European calendar.  Al-Sharh al-Mumti’, 6/471 
May Allah make your missing out on this special reward, and the missing out of others who are keen like you, lead to a good outcome, so this will make you strive harder in doing good works and acts of worship. This may make you more keen than people who do some act of worship regularly and feel content with what they are doing, and may fill them with self-admiration. 
We ask Allah to bless us with His bounty and reward, and help us to remember Him and give thanks to Him.