Asha akirudi nyumbani, nitamwonyesha kitabu changu kipya.

It is made of three parts:

• a- = she/he subject marker
• -ki- = a marker often used for if/when
• -rudi = return / come back

So Asha akirudi literally means something like Asha, if/when she returns.

It can mean either, depending on context.

In this sentence, when is usually the best translation because Asha’s return sounds expected:
When Asha comes back home, I will show her my new book.

But grammatically, if is also possible if the return is uncertain.

Because in Swahili, the subject marker on the verb is normally still used even when the full noun is present.

So:

• Asha akirudi = Asha, when she returns
• not just Asha kirudi

The name Asha identifies who we are talking about, and a- is still required as part of normal verb grammar.

Nyumbani means home / at home.

It comes from:

• nyumba = house
• -ni = a locative ending, giving the sense of in/at/to

So nyumbani is a locative form meaning something like at home or home.
With motion verbs, Swahili often does not need a separate preposition where English might use to.

So akirudi nyumbani = when she returns home.

It breaks down like this:

• ni- = I
• -ta- = future marker, will
• -mw- = her/him
• -onyesha = show

So nitamwonyesha means I will show her/him.

In this sentence, it means I will show her, because the person is Asha.

Swahili commonly puts object pronouns inside the verb as object markers.

So instead of a separate word for her, Swahili uses -mw- inside nitamwonyesha.

That is very normal Swahili structure:

• ni-ta-mw-onyesha
• I-will-her-show

The object marker for a singular person, such as him/her, is basically m-.
Before a vowel-initial verb stem like -onyesha, it appears as mw- for easier pronunciation.

So:

• m- + onyesha becomes mwonyesha

That is why the form is nitamwonyesha.

Yes, and that is normal with a verb like -onyesha.

In this sentence:

• -mw- = the person being shown something, her
• kitabu changu kipya = the thing being shown, my new book

So Swahili can express both:

• who you are showing it to
• what you are showing

all in one sentence.

In Swahili, the noun usually comes first, and modifiers follow it.

So:

• kitabu = book
• changu = my
• kipya = new

That gives kitabu changu kipya = my new book.

The important thing for a learner is:

• the noun comes first
• the words after it must agree with the noun class of kitabu

Because possessives in Swahili must agree with the noun class of the thing possessed.

Kitabu belongs to noun class 7 singular, so my takes the class-7 possessive form changu.

So:

• kitabu changu = my book

If you used yangu, that would not agree correctly with kitabu.

Because adjectives also agree with the noun class.

The adjective stem is -pya = new.
With kitabu (class 7 singular), it becomes kipya.

So:

• kitabu kipya = new book

The ki- shows agreement with kitabu.

Because they belong to different types of modifier.

• changu is a possessive
• kipya is an adjective

Different modifier types use different agreement patterns.
For class 7 singular:

• possessive agreement gives ch- → changu
• adjective agreement gives ki- → kipya

So both agree with kitabu, but they do not have to look identical.

The comma helps show that the first part is a subordinate clause:

• Asha akirudi nyumbani,
• nitamwonyesha kitabu changu kipya.

It is similar to English When Asha comes home, ...
In writing, the comma makes the structure clearer. In speech, there would usually be a small pause.