Этот мост построили в прошлом веке.

Because построили does not agree with мост here.

This sentence uses an indefinite-personal construction: the subject is omitted, but the plural past verb implies they / people / someone.

So the literal structure is closer to:

• Этот мост построили... = They built this bridge...

The bridge is the object, not the subject.

Here it is accusative singular: it is the thing that was built.

The tricky part is that for masculine inanimate nouns, the accusative looks exactly like the nominative:

• этот мост = nominative
• этот мост = accusative

So the form itself does not show the difference; the verb and sentence structure do.

Compare with an animate noun, where the forms differ:

• nominative: Этот человек
• accusative: Этого человека

That is why этот мост can look confusing at first.

Russian often leaves the subject out when it is unknown, unimportant, or general.

So instead of saying:

• Они построили этот мост... = They built this bridge...

Russian can simply say:

• Этот мост построили...

This does not usually mean specific known they. It often means something like:

• they
• people
• someone
• it was done

depending on context.

Because Russian often prefers an active indefinite-personal construction where English prefers a passive.

So:

• Этот мост построили в прошлом веке. literally: They built this bridge in the last century.

But natural English is often:

• This bridge was built in the last century.

Russian can also use a true passive:

• Этот мост был построен в прошлом веке.

That version is closer to the English passive and can sound a bit more formal or written.

Because построили is perfective, and it presents the bridge as a completed result.

• строили = were building / used to build / built, with focus on the process
• построили = built, finished building

Since the sentence is about the fact that the bridge came into existence in that century, perfective is the natural choice.

It is the prepositional case.

After в meaning in with a time period like this, Russian uses the prepositional:

• в веке = in the century
• в прошлом веке = in the last / previous century

The adjective agrees with the noun:

• прошлый век = nominative
• в прошлом веке = prepositional

Because adjectives must agree with the noun in gender, number, and case.

Here the noun is:

• век — masculine singular

And after в in this expression, it is in the prepositional case:

• в веке

So the adjective also changes to prepositional masculine singular:

• прошлый → прошлом

Hence:

• в прошлом веке

In normal English translation, it usually means in the last century or in the previous century.

Literally, прошлый means past / previous / last, but in this expression the natural meaning is:

• in the previous century

In modern everyday context, that often means the 20th century, but it depends on the time frame of the speaker or text.

Yes. Russian word order is flexible, and changing it can shift the focus or topic.

For example:

• Этот мост построили в прошлом веке.
  The bridge is the starting point/topic.

• В прошлом веке построили этот мост.
  The time phrase is emphasized more.

• Этот мост в прошлом веке построили.
  Possible, with stronger emphasis or contrast.

So the sentence order here is natural, but not the only option.

The basic meaning is the same, but the style is a bit different.

• Этот мост построили в прошлом веке.
  More natural conversational Russian; active form with unspecified people.

• Этот мост был построен в прошлом веке.
  More explicitly passive; often sounds more formal, written, or factual.

Both are correct.

Russian has no articles.

So Russian does not have separate words for a or the. Definiteness is understood from context, word order, or words like этот.

Here этот already makes the noun definite:

• этот мост = this bridge

So no article is needed.