Мне не удалось понять это правило с первого раза.

Because this sentence uses an impersonal construction:

кому-то удалось + infinitive = someone managed to do something

In this pattern, the person is marked with the dative:

• мне удалось = I managed
• ему удалось = he managed
• нам удалось = we managed

So мне does not mean to me word-for-word in normal English style; it marks the person for whom the success happened.

Удалось is the past tense of удаться, a verb meaning something like:

• to succeed
• to work out
• to turn out successfully
• in this construction, to manage

So:

• Мне удалось понять = I managed to understand
• Мне не удалось понять = I didn't manage to understand / I failed to understand

A very literal sense is something like “It did not succeed for me to understand…”, which explains why the sentence is built differently from English.

Because не удалось понять focuses on failure to achieve the result, not simply on the fact that understanding did not happen.

Compare:

• Я не понял это правило.
  = I didn't understand this rule.

• Мне не удалось понять это правило.
  = I didn't manage to understand this rule.

The second version sounds a bit more like:

• I tried
• I made an effort
• but I still couldn't get it

So it often feels a little more natural when talking about difficulty or unsuccessful effort.

Here удалось is in the neuter singular past, which is the normal form in many impersonal expressions in Russian.

There is no grammatical subject like я controlling the verb. The construction is impersonal, so Russian uses the default neuter singular past form:

• мне удалось
• ему удалось
• ей удалось
• нам удалось

So удалось does not agree with мне. It stays in that impersonal form.

Понять is the perfective verb, and it refers to reaching the result of understanding.

In this sentence, the idea is not just “I was in the process of understanding,” but rather “I failed to achieve understanding.” That is why Russian normally uses понять here:

• удалось понять = managed to understand
• не удалось понять = didn't manage to understand

Using понимать would sound unusual here, because удаться / не удаться usually combines with an action understood as a complete achievement.

Because правило is a neuter noun.

The demonstrative этот changes by gender:

• этот = masculine
• эта = feminine
• это = neuter

So:

• это правило = this rule

Also, правило is in the accusative, but for an inanimate neuter noun, the accusative looks the same as the nominative, so the form stays это правило.

It is in the object case: accusative.

The verb понять takes a direct object, so правило is accusative. But with inanimate neuter nouns, the accusative form is the same as the nominative.

So:

• nominative: это правило
• accusative: это правило

They look identical, even though the grammatical role is different.

С первого раза is a very common Russian expression meaning:

• on the first try
• the first time
• at the first attempt

So the sentence means that the speaker did not understand the rule immediately / on the first attempt.

This phrase is idiomatic and very common in everyday Russian:

• Я не запомнил это с первого раза. = I didn't remember it the first time.
• У неё получилось с первого раза. = She got it on the first try.

This is just the standard idiomatic pattern.

After с, Russian often uses the genitive, and in the expression с первого раза the noun раз becomes раза.

So:

• с первого раза = from the first attempt / on the first try

You should mostly learn this as a fixed expression rather than trying to translate it word for word.

A useful contrast:

• в первый раз = for the first time
• с первого раза = on the first try

These are close in meaning, but с первого раза emphasizes success or failure on an attempt.

Yes, absolutely. That is a very natural alternative.

Compare:

• Мне не удалось понять это правило с первого раза.
• Я не смог понять это правило с первого раза.

Both mean roughly the same thing, but there is a slight nuance:

• Я не смог... = more direct, personal: I couldn't...
• Мне не удалось... = slightly less direct, often softer, more like I didn't manage to...

The version with удалось can sound a bit more formal or neutral, while не смог is very common in everyday speech.

Не negates удалось:

• Мне удалось понять = I managed to understand
• Мне не удалось понять = I did not manage to understand

So the negation applies to the idea of success/managing, not directly to understand.

That is why the meaning is not just I didn’t understand, but more specifically I failed to understand.

No, Russian word order is flexible. The original sentence is natural and neutral:

• Мне не удалось понять это правило с первого раза.

But other orders are possible, depending on emphasis:

• Это правило мне не удалось понять с первого раза.
  Emphasizes this rule.

• С первого раза мне не удалось понять это правило.
  Emphasizes on the first try.

The original order is a very normal way to present the whole idea smoothly.

It sounds natural, standard, and slightly more polished than the simplest conversational version.

• Мне не удалось понять это правило с первого раза is perfectly normal and educated-sounding.
• A more everyday, simpler version might be Я не смог понять это правило с первого раза.
• An even more direct version is Я не понял это правило с первого раза.

So the original sentence is not overly formal, but it does sound a little more structured than the most basic spoken alternative.