Сестра попросила меня занести документы в офис и забежать в аптеку по дороге.

Попросила is the past tense feminine singular form of попросить.

The ending -ла shows that the subject is:

• past tense
• singular
• feminine

So because the subject is сестра, the verb becomes попросила.

Compare:

• брат попросил
• сестра попросила
• они попросили

This is a question of aspect.

• попросила = perfective: she made the request as a completed single action
• просила = imperfective: she was asking, used to ask, or the request is viewed as a process

In this sentence, the natural meaning is that the sister made one specific request, so попросила is the normal choice.

After попросить, the person who is being asked usually goes in the accusative case.

The pattern is:

попросить кого? + infinitive

So:

• попросила меня = asked me
• попросила брата = asked her brother
• попросила друзей = asked her friends

Even though English says asked me to..., Russian does not use the dative here.

After попросить кого-то, Russian normally uses an infinitive to say what the person was asked to do.

Pattern:

• попросить кого-то сделать что-то

So:

• попросила меня занести документы
• попросила меня забежать в аптеку

Both infinitives depend on попросила. The understood subject of both actions is меня: I am the one who is supposed to bring the documents and stop by the pharmacy.

Because these are seen as single, completed errands.

• занести = to drop off / bring in once, successfully
• забежать = to pop in once

The sister is asking for two specific completed actions:

1. deliver the documents
2. stop by the pharmacy

If you used imperfective forms such as заносить or забегать, it would sound more like repeated action, habit, or focus on process, which does not fit this sentence well.

Here занести means something like:

• drop off the documents
• bring the documents in/to the office

It is a very natural verb for taking something to a place as part of an errand.

It is related to motion/carrying, but in real usage it often has the practical sense of bringing something somewhere and leaving it there.

A rough comparison:

• отнести документы = take the documents there
• принести документы = bring the documents here
• занести документы = bring/drop them off at a place, often as a brief stop

Because the sentence describes motion toward a destination, not location.

With в:

• в + accusative = into / to
• в + prepositional = in / inside

So:

• в офис = to the office
• в аптеку = to the pharmacy
• в офисе = in the office
• в аптеке = in the pharmacy

Since the speaker is talking about going somewhere, Russian uses the accusative.

Literally, yes, забежать is connected with running in. But in everyday Russian it often means:

• stop by
• pop into
• drop in briefly

So here it does not necessarily mean physical running. It usually suggests a quick visit, which fits well with an errand done on the way.

По дороге means on the way.

It is a very common expression. Here дороге is the dative singular of дорога, because the preposition по often takes the dative.

So:

• дорога → дороге

In this phrase, the meaning is not strictly about a literal road. It often means:

• while going somewhere
• along the way
• during the trip from one place to another

So в аптеку по дороге means something like stop by the pharmacy on the way.

Yes, that is possible.

Russian has two common ways to express this idea:

1. попросила меня + infinitive

   • more compact
   • very common
   • especially natural when the person asked is clear

2. попросила, чтобы я + past tense verb

   • more explicit
   • can add emphasis
   • useful when the clause is longer or more detailed

So you could say something like:

Сестра попросила, чтобы я занёс документы в офис и забежал в аптеку по дороге.

That is grammatical, but the original version with infinitives is more concise and very natural.