Dought & clear, - Will the person who intends to fast ‘Ashoora during the day be rewarded for his fasting?

I know of the virtue of fasting the day of 'Ashoora, and that it is
expiation for the year that came before. But because we use the
Gregorian calendar I did not know when 'Ashoora 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 Allaah.
Praise be to Allaah for making you keen to do naafil 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 Allaah be upon him) that the intention to observe a
naafil 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 Allaah be pleased with her) narrated that the
Prophet(peace and blessings of Allaah 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 naafil 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 inSaheeh 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 Allaah 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 Allaah 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, 14thand 15thof 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
'Ashoora until after dawn had come: he will not get the reward for
fasting 'Ashoora', which is the expiation of one year, because his
case is not like that of one fasted the whole day of 'Ashoora', 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 Allaah Muharram, which is the best kind of fasting
after Ramadaan (as it says inSaheeh Muslim, 1163).
Perhaps one of the main reasons for your and many people's not knowing
when 'Ashoora is, oral-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 Allaah blesses
with guidance, to follow the lunar Hijri calendar which Allaah 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 'Ashoora – which
is a day that is known through the lunar calendar. They said that it
was the day on which Allaah 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 Allaah 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 Allaah to bless us with His bounty and reward, and help us to
remember Him and give thanks to Him.