Learn Polish: Tamtej nocy przeczytałem pierwszy wpis. - That night I read the first entry.

This is the bare Genitive of time taught in this lesson. Polish uses the Genitive case without a preposition to indicate a specific, definite point in time, usually when narrating past events. You will often see this with phrases like tamtej nocy (that night), następnego dnia (the next day), or pewnego razu (once upon a time).

Would it be wrong to say w tamtą noc instead of tamtej nocy?

W tamtą noc (using the preposition w plus the Accusative case) is grammatically correct and means the same thing. However, the bare Genitive tamtej nocy is much more natural and stylistic for storytelling or narrating a sequence of past events.

Could I use tej nocy instead of tamtej nocy here?

You could, but it subtly changes the timeline. Tej nocy means this night (referring to the night that just passed, or the current one). Because the sentence is looking back at a more distant, specific time in a narrative, tamtej nocy (that night) is the better fit.

Why does the direct object pierwszy wpis look exactly like the dictionary form? Shouldn't the Accusative case change it?

Wpis (entry or post) is an inanimate masculine noun. As a quick refresher: for inanimate masculine nouns, the Accusative case is identical to the Nominative case. If it were a feminine noun or an animate masculine noun, the endings would have changed.