मैं घर आऊँगी, फिर आपको फोन करूँगी।

That ending marks the future tense for first-person singular feminine in this sentence.

• आऊँगी = I will come (said by a female speaker)
• करूँगी = I will do / I will make → here, फ़ोन करूँगी = I will call

A useful comparison:

• masculine speaker: आऊँगा, करूँगा
• feminine speaker: आऊँगी, करूँगी

So the ending is telling you both:

1. the action is in the future
2. the speaker is female

Yes. The forms आऊँगी and करूँगी are feminine singular.

If a male speaker said the same sentence, it would be:

मैं घर आऊँगा, फिर आपको फोन करूँगा।

In Hindi, many verb forms agree with the speaker’s or subject’s gender, so this is something learners need to watch carefully.

Because घर आना is a very common expression meaning to come home.

In Hindi, destination words are often used without a postposition in set expressions like this:

• घर जाना = to go home
• घर आना = to come home

Compare:

• घर आऊँगी = I’ll come home
• घर में आऊँगी = I’ll come into the house
  This sounds different and focuses on entering the house.
• घर को आऊँगी is not natural here.

So घर by itself is correct and idiomatic.

Good question. Both आना (to come) and जाना (to go) can be used with घर, but they reflect slightly different viewpoints.

• घर आऊँगी = I’ll come home
• घर जाऊँगी = I’ll go home

Hindi, like English, sometimes uses come home very naturally, especially when the idea is returning to one’s own place. So घर आऊँगी is perfectly normal.

In some situations, घर जाऊँगी would also be possible. The choice depends on how the speaker is framing the movement.

Here फिर means then or after that.

So the sequence is:

1. मैं घर आऊँगी = I’ll come home
2. फिर आपको फोन करूँगी = then I’ll call you

Be careful: फिर can also mean again in other contexts.

For example:

• मैं फिर आऊँगी। = I’ll come again.

In your sentence, because it connects two actions in order, it means then, not again.

Because the person being called is marked with को.

With फ़ोन करना or कॉल करना, Hindi often uses the person called as an indirect/direct object with को:

• मैं आपको फोन करूँगी। = I’ll call you.
• मैं उसे फोन करूँगी। = I’ll call him/her.
• मैं तुम्हें फोन करूँगी। = I’ll call you. (less formal)

Also remember:

• आप is the formal/polite you
• आपको is to you / you in an object role

So आपको is exactly what you expect here.

Yes, it’s completely normal.

Hindi very often combines a borrowed noun with करना to make a verb phrase. So:

• फोन करना = to call / to phone
• मैसेज करना = to message
• कॉल करना = to call

Literally, it may look like do a phone, but idiomatically it just means to phone/call.

So आपको फोन करूँगी is a standard, natural way to say I’ll call you.

Because once the subject is clear, Hindi often leaves it out in the next clause.

So this is natural:

• मैं घर आऊँगी, फिर आपको फोन करूँगी।

A more explicit version would be:

• मैं घर आऊँगी, फिर मैं आपको फोन करूँगी।

That is grammatical, but usually less natural unless you want extra emphasis on I.

Hindi often drops repeated subjects when they are already understood from context.

The mark ँ shows nasalization of the vowel.

So:

• आऊँगी is pronounced roughly aa-oon-gee, with a nasal sound on ऊँ
• करूँगी is roughly kuh-roon-gee, again with nasalization

It is not a full separate n sound like in English soon + g. It’s more like the vowel itself becomes nasal.

If you are reading carefully:

• आऊँगी = आ + ऊँ + गी
• करूँगी = कर + ऊँ + गी

This nasalized future ending is very common, so it’s worth getting used to it.

Yes, Hindi word order is somewhat flexible, but the given order is very natural.

Your sentence:

मैं घर आऊँगी, फिर आपको फोन करूँगी।

works well because it clearly presents:

1. the first action
2. the second action

You may hear slight variations, but moving words around can change the emphasis.

For example:

• मैं फिर आपको फोन करूँगी। can mean I’ll call you again
• फिर right before the second clause strongly gives the sense then / after that

So the original sentence is a very good, standard way to say it.

Both are natural, but they are structured a bit differently.

1. मैं घर आऊँगी, फिर आपको फोन करूँगी।

   • two separate future actions
   • I’ll come home, then I’ll call you

2. मैं घर आकर आपको फोन करूँगी।

   • literally having come home, I’ll call you
   • more like I’ll call you once I get home / after coming home

The version with आकर links the actions more tightly.
The version in your sentence presents them more clearly as a sequence: first this, then that.

Both are useful, and learners will hear both kinds of structure often.