Pulma ratkesi lopulta.

Because these are two different verbs:

• ratketa = to be solved, to come undone, to resolve itself
• ratkaista = to solve something

So:

• Pulma ratkesi lopulta. = the problem got solved / was solved in the end
• Joku ratkaisi pulman lopulta. = someone solved the problem in the end

This is a very common Finnish distinction: one verb is intransitive and describes something happening, while the other is transitive and takes an object.

Pulma is nominative singular.

In this sentence it is the subject of the verb:

• pulma = the problem / puzzle
• ratkesi = got solved
• lopulta = finally / in the end

So pulma is the thing that underwent the change.

Because ratketa is an intransitive verb here. It does not take an object in this meaning.

Compare:

• Pulma ratkesi.
  The problem got solved.
  → no object

• Hän ratkaisi pulman.
  He/she solved the problem.
  → pulman is the object

That is why you see pulma in the nominative, not an object form.

Because Finnish often expresses this idea with an intransitive verb rather than a passive.

So Pulma ratkesi is literally closer to:

• The problem resolved
• The problem got solved

Even though English often prefers was solved, Finnish does not need a passive here. It is a natural Finnish way to describe the situation.

It does not necessarily mean the problem magically solved itself; it just means that, in the end, the problem ended up solved.

Lopulta is an adverb meaning:

• finally
• eventually
• in the end

In this sentence it tells you when or after what process the solving happened: it happened after some delay or difficulty.

So the sentence suggests that the problem was not solved immediately; it took some time, but it was solved in the end.

Yes. Finnish word order is fairly flexible.

The neutral order here is:

• Pulma ratkesi lopulta.

But you could also say:

• Lopulta pulma ratkesi.
• Pulma lopulta ratkesi.

These all mean roughly the same thing, but the focus changes a little:

• Pulma ratkesi lopulta. = neutral
• Lopulta pulma ratkesi. = stronger emphasis on finally / in the end
• Pulma lopulta ratkesi. = slight emphasis on the process ending in resolution

So the original sentence is a normal, natural basic order.

It is the simple past tense, often called the imperfect in Finnish grammar.

So:

• ratkeaa = gets solved / is getting solved
• ratkesi = got solved / was solved

This sentence is talking about a completed event in the past.

This is something learners often have to get used to: Finnish verbs do not always look very transparent once they are conjugated.

Here, the dictionary form is ratketa, but the past form is ratkesi. The stem changes, so this is best learned as a pattern or as a pair:

• ratketa → ratkesi

You do not need to force a word-for-word breakdown every time. At first, it is perfectly normal to simply learn:

• ratketa = to get solved
• ratkesi = got solved

Yes. Finnish has no articles, so pulma can correspond to:

• a problem
• the problem

Which one sounds best depends on context.

Without context, English often chooses the problem if both speaker and listener already know which problem is being discussed. But Finnish itself does not mark that difference with an article.

You would use the transitive verb ratkaista and put problem in the object form:

• Joku ratkaisi pulman lopulta.

Here:

• joku = someone
• ratkaisi = solved
• pulman = the problem, as the object
• lopulta = finally / in the end

So the contrast is:

• Pulma ratkesi lopulta.
  The problem got solved in the end.

• Joku ratkaisi pulman lopulta.
  Someone solved the problem in the end.