Learn Hardcore Latin: Arvum post villam est. - The ploughed field is behind the villa.

Because arvum is a neuter noun of the second declension. In Latin, many second-declension neuter nouns have -um in the nominative singular, which is the case normally used for the subject.

So even though an English speaker might not expect a subject to look like arvum, that form is completely normal for a neuter subject in Latin.

Why is it villam and not villa?

Because post takes the accusative case. The basic form of the noun is villa, but after post it must become villam.

So:

villa = the dictionary form
villam = accusative singular, used after post

Does post mean after or behind here?

It can mean either one, depending on context.

In a time expression, post often means after
In a place expression, post often means behind

Here, since the sentence is telling you where something is, post villam is understood spatially: behind the house/farmhouse