हम समय पर शहर पहुँचेंगे, और वहाँ माँ से मिलेंगे।

समय पर means on time. Literally, पर is a postposition meaning on/upon, so Hindi expresses this idea as something like on time rather than using a single adverb.

This is also a good example of how Hindi uses postpositions after nouns, where English uses prepositions before nouns.

With the verb पहुँचना meaning to reach/arrive, Hindi often uses the destination directly: घर पहुँचना, स्कूल पहुँचना, शहर पहुँचना.

So शहर पहुँचेंगे is a normal way to say will reach the city.
Using में would change the feel a bit, making it more like being located in the city rather than simply reaching it. को is not used here.

पहुँचेंगे is the future tense form of पहुँचना. The ending -ेंगे agrees with हम.

In this sentence, it is the plural future form, and it is also the usual masculine/mixed-gender form. If the speakers were all female, you would normally use पहुँचेंगी instead.

Because this sentence is in the future, not a perfective past construction.

The marker ने is mainly used with many transitive verbs in perfective clauses, such as certain past-tense sentences. It is not used in a simple future sentence like this one. So हम पहुँचेंगे is correct, not हमने पहुँचेंगे.

और means and. It joins the two actions:

• हम समय पर शहर पहुँचेंगे
• वहाँ माँ से मिलेंगे

So the sentence is saying that both future actions will happen: first reaching the city, and then meeting mother there.

वहाँ means there. It refers back to the city mentioned earlier.

It helps link the second action to that location: we will meet mother there.
You could leave it out if the place were already obvious from context, but including वहाँ is very natural.

Because the verb मिलना meaning to meet normally takes से with the person you meet.

So माँ से मिलेंगे is the standard pattern. It is close to the idea of meet with mother.

Using को here would be incorrect for this meaning.

Because the subject is already clear from the first clause. Hindi often leaves out a repeated subject when it is understood.

So:

• हम समय पर शहर पहुँचेंगे, और वहाँ माँ से मिलेंगे।

is perfectly natural.

You could repeat हम for emphasis, but it is not necessary.

Hindi often leaves out possessive words with close family terms when the meaning is clear from context. So माँ here can simply mean mother/mom, and the exact relationship is understood from the situation.

If you wanted to make it explicit, you could say:

• हमारी माँ = our mother
• अपनी माँ = our own mother

Yes, Hindi word order is somewhat flexible, but the most neutral order usually keeps the main verb near the end.

This sentence uses a very natural order:

• हम = subject
• समय पर = time phrase
• शहर = destination
• पहुँचेंगे = verb

Then the second clause follows the same general pattern.

You can move some parts for emphasis, but not every rearrangement sounds equally natural. For a learner, the given order is a very good model to follow.