Мы закончили этот проект третьего апреля две тысячи третьего года.

Breakdown of Мы закончили этот проект третьего апреля две тысячи третьего года.

мы
we
этот
this [f: эта, n: это]
год
year
две
two (f)
проект
the project
апрель
April
тысяча
thousand
третий
third
закончить
to finish (perfective)

Questions & Answers about Мы закончили этот проект третьего апреля две тысячи третьего года.

Why do we use the genitive case for the date (третьего апреля ... третьего года)?
When expressing that an event happened on a specific full date (day, month, and year), Russian uses the genitive case for the entire date expression. You don't need a preposition like "on", the genitive case alone carries that meaning.
What gender is the ordinal number for the day (третьего), and why?
It is neuter. The word for the day of the month is an ordinal number that technically describes the hidden neuter word число (date or number). So, третьего is the genitive form of the neuter word третье.
In the year (две тысячи третьего), why does only the last word change to the genitive case?
This is a strict rule for compound ordinal numbers in Russian. When you have a complex number acting as an ordinal (like the two thousand and third year), only the very last word takes the ordinal endings and changes case. The earlier parts, like две тысячи, stay as normal cardinal numbers in their base form.
Since этот проект is the direct object being finished, shouldn't it change its ending for the accusative case?
It is in the accusative case! Because проект is a masculine inanimate noun (not a living person or animal), its accusative form looks exactly the same as its dictionary form. The pronoun этот follows the exact same rule.

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