После обновления программа снова заработала правильно.

Because после requires the genitive case.

The base noun is обновление (update, updating).
Its genitive singular form is обновления.

So:

• обновление = nominative
• после обновления = after the update / after updating

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

It can mean either, depending on context.

Обновление is a verbal noun, so it may refer to:

• the update itself
• the process of updating

In this sentence, После обновления программа снова заработала правильно, English would usually understand it as After the update, the program started working properly again.

So here обновления most naturally means the update.

Because заработала does not simply mean worked. It usually means started to work, began functioning, or came back into operation.

Compare:

• программа работала правильно = the program was working properly
• программа заработала правильно = the program started working properly / began to function properly

The prefix за- often gives this starting meaning to verbs:

• петь = to sing

• запеть = to start singing

• говорить = to speak
• заговорить = to start speaking

So заработала suggests that before this point, the program was not functioning correctly, and then it began functioning correctly.

Because the subject, программа, is feminine singular.

In the past tense, Russian verbs agree with the subject in gender and number:

• masculine: заработал
• feminine: заработала
• neuter: заработало
• plural: заработали

Since программа is feminine, the correct form is заработала.

Снова means again.

In this sentence, it tells us that the program had worked properly before, then stopped working properly, and after the update it worked properly again.

So the sense is:

• it was okay before
• something went wrong
• after the update, it became okay again

A close synonym is опять. In many everyday contexts, снова and опять are interchangeable.

That is a natural and common position, but Russian word order is more flexible than English word order.

Here:

• После обновления = time/background
• программа = subject
• снова = again
• заработала = started working
• правильно = properly

So the sentence flows naturally as:

После обновления программа снова заработала правильно.

You could move words around for emphasis, for example:

• После обновления программа заработала правильно снова — possible in theory, but much less natural
• После обновления снова программа заработала правильно — possible, but marked and unusual

The original order is the most neutral.

Here правильно means properly, correctly, or as it should.

It does not mean something like right in a moral sense. It means the program functioned in the correct way.

So:

• работать правильно = to work properly
• работать неправильно = to work incorrectly / malfunction

With software or machines, правильно often means without errors, normally, or as expected.

Most naturally, it means started working properly again.

That comes from заработала, which usually implies the beginning of functioning. So the idea is not just a state, but a change:

• before: the program was not working properly
• after the update: it began working properly again

If you wanted to describe an ongoing state more simply, you might use работала in a different context. But in this sentence, заработала strongly suggests a return to functioning.

Because Russian has no articles like a or the.

Russian usually leaves definiteness to context. So:

• программа can mean a program or the program
• обновление can mean an update or the update

In this sentence, the context makes them feel definite in English, so the natural translation uses the:

• the update
• the program

But Russian does not need separate words for that.

Yes, that is perfectly natural.

Снова and опять both often mean again. In this sentence, either works:

• После обновления программа снова заработала правильно.
• После обновления программа опять заработала правильно.

A rough tendency:

• снова can sound a little more neutral or bookish
• опять can sound a little more conversational

But in many real situations, the difference is small.

Yes, заработать can also mean to earn.

For example:

• заработать деньги = to earn money

But Russian verbs often have more than one meaning, and context tells you which one is intended.

Here the subject is программа (program/software), so заработала can only realistically mean started working / began functioning, not earned anything.

So with people, заработать may mean earn.
With machines, systems, software, or devices, it often means start working.

They are similar in meaning, but not identical in style.

• После обновления = after the update / after updating

  • shorter
  • more compact
  • very natural in written and spoken Russian

• После того, как обновили программу = after they updated the program / after the program was updated

  • longer
  • more explicit
  • uses a clause instead of a noun phrase

Russian often prefers these compact noun-based expressions, especially in technical or neutral statements.

The main stress is:

• послЕ
• обновлЕния
• прогрАмма
• снОва
• заработАла
• прАвильно

A rough pronunciation guide:

• После ≈ POS-lye
• обновления ≈ ab-nav-LYE-ni-ya
• программа ≈ pra-GRAM-ma
• снова ≈ SNO-va
• заработала ≈ za-ra-BO-ta-la
• правильно ≈ PRA-vil-na

In normal speech, the sentence is usually said smoothly, with the main information focus often falling on заработала правильно or снова заработала правильно, depending on what the speaker wants to emphasize.