Ливень начался так внезапно, что я не успел открыть зонт.

Ливень means a downpour or heavy rain, not just ordinary rain.
So this sentence is talking about rain that started very strongly and suddenly.

• дождь = rain
• ливень = a heavy shower / downpour

Using ливень makes the sentence more vivid.

Начался is the perfective past form of начаться, meaning the event started as a complete action.

In this sentence, the speaker is focused on the moment the downpour began:

• начался = it started
• начинался = it was starting / used to start / was beginning

Because the sentence describes one sudden completed event, начался is the natural choice.

The base verb is начаться, not начать.

• начать = to begin something
• начаться = to begin, to start

So:

• Я начал урок. = I began the lesson.
• Урок начался. = The lesson began.

In your sentence, the rain itself is the thing that began, so Russian uses начаться.

Because ливень is a masculine singular noun.

Past tense verbs in Russian agree with gender and number:

• masculine: начался
• feminine: началась
• neuter: началось
• plural: начались

Since ливень is masculine, the verb must be начался.

This is a very common Russian pattern meaning so ... that ...

In your sentence:

• так внезапно = so suddenly
• что я не успел открыть зонт = that I didn’t have time to open my umbrella

So the structure is:

• так + adverb/adjective + что + result

Examples:

• Он говорил так тихо, что я ничего не слышал. = He spoke so quietly that I heard nothing.
• Было так холодно, что мы ушли домой. = It was so cold that we went home.

Because it describes how the rain started, so Russian needs an adverb.

• внезапный = sudden (adjective)
• внезапно = suddenly (adverb)

Compare:

• внезапный ливень = a sudden downpour
• ливень начался внезапно = the downpour started suddenly

In this sentence, we are modifying the verb начался, so внезапно is correct.

Не успел means didn’t manage to in time, didn’t have time to, or wasn’t quick enough to.

The verb is успеть, which often expresses managing to do something before it is too late.

So:

• Я успел открыть зонт. = I managed to open the umbrella in time.
• Я не успел открыть зонт. = I didn’t manage to open the umbrella in time.

This is slightly more specific than simply I didn’t open the umbrella. It suggests the rain began too fast.

After успеть / не успеть, Russian usually uses the infinitive that fits the result being discussed.

Here, открыть is perfective, meaning to open as a completed action.

• не успел открыть зонт = I didn’t manage to open the umbrella
• не успел открывать would sound wrong here in normal usage

The point is whether the speaker managed to complete the action before the rain came down, so perfective открыть is the natural choice.

Because it is the direct object of открыть, so it is in the accusative case.

For many inanimate masculine nouns, the accusative looks exactly like the nominative:

• nominative: зонт
• accusative: зонт

So even though the case has changed grammatically, the form stays the same.

Compare with a feminine noun:

• открыть дверь = to open the door
  Here дверь is also accusative, but the form happens to look the same as nominative too.

Russian often leaves out possessive words like my, your, his when they are obvious from context.

In English, we normally say open my umbrella.
In Russian, открыть зонт is often enough if it is clear whose umbrella it is.

You could say открыть свой зонт, but that is not necessary here and may sound heavier than needed.

So:

• открыть зонт = open the umbrella / my umbrella, depending on context
• открыть свой зонт = open one’s own umbrella, more explicit

Yes. Russian word order is flexible, though the original sentence is the most neutral and natural.

Original:

• Ливень начался так внезапно, что я не успел открыть зонт.

You could also hear:

• Так внезапно начался ливень, что я не успел открыть зонт.

This version gives a bit more emphasis to так внезапно.

Russian word order often changes for emphasis, rhythm, or style, but the basic meaning stays the same because the grammar shows the relationships between the words.

It is not always strictly necessary, because Russian verbs often show the person clearly. But here я sounds natural and helps make the clause clear.

You could say:

• ...что не успел открыть зонт.

That is possible in conversation, especially if the subject is already obvious.

Still, keeping я is very normal and often better for learners to model.

A very natural English rendering would be:

• The downpour started so suddenly that I didn’t have time to open my umbrella.
• It started raining so suddenly that I didn’t manage to open my umbrella in time.

This helps show an important nuance: не успел is not just didn’t open; it carries the idea of not having enough time.