Mama akifika nyumbani, nitamwonyesha shajara yangu binafsi.

Akifika breaks down as:

• a- = he/she
• -ki- = if/when/as
• -fika = arrive

So akifika means if/when she arrives here. Since the subject is Mama, the meaning is when Mom arrives.
A useful thing to remember: Swahili uses the same subject marker a- for both he and she.

Because Swahili often puts that meaning inside the verb itself. The marker -ki- can mean:

• when
• if
• as/while

In this sentence, because the second clause is about a future action (nitamwonyesha = I will show her), akifika is most naturally understood as when she arrives.

Yes. That is also correct.

There is a nuance:

• Mama akifika nyumbani... = when/if Mom arrives home...
• Mama atakapofika nyumbani... = more explicitly when Mom arrives home in the future

So atakapofika sounds a bit more clearly future and definite.
Akifika is shorter and very common, but it can sometimes also mean if she arrives, depending on context.

Nyumba means house/home.
Adding -ni makes it locative:

• nyumba = house, home
• nyumbani = at home, home, to home

After a verb like fika (arrive), nyumbani means home or at home. So akifika nyumbani is when she arrives home.

It breaks down like this:

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

So nitamwonyesha means I will show him/her.
In this sentence, mw- refers to Mama, so the meaning is I will show her.

Because they refer to different roles in the sentence.

• mw- = the person being shown something, so her
• shajara yangu binafsi = the thing being shown, my personal diary

So the structure is basically:

• nitamwonyesha = I will show her
• shajara yangu binafsi = my personal diary

Together: I will show her my personal diary.

Because possessives in Swahili must agree with the noun class of the noun they describe.

Shajara belongs to the N/N class (often called class 9/10), so it takes ya- type possessives:

• yangu = my
• yako = your
• yake = his/her

That is why we say shajara yangu, not shajara changu.

For comparison:

• kitabu changu = my book
  because kitabu is a different noun class

In Swahili, possessives usually come after the noun, not before it.

So:

• shajara yangu = literally diary my
• natural English = my diary

This is normal Swahili word order. The same is true for many other modifiers as well: the noun often comes first.

Binafsi means personal, private, or sometimes one’s own depending on context.

So:

• shajara yangu = my diary
• shajara yangu binafsi = my personal/private diary

It adds emphasis that this diary is personal, not just any diary that belongs to me.

Yes, very often.

In Swahili, kinship terms like Mama, Baba, Kaka, and Dada are often used without a possessive when the relationship is clear from context. So Mama can naturally mean:

• Mom
• Mother
• sometimes simply mum

If you wanted to be more explicit, you could also say mama yangu.

Kuonyesha is the infinitive form, meaning to show.

When a verb is conjugated in a sentence, the infinitive prefix ku- is removed, and other markers are added instead. So:

• kuonyesha = to show
• ni-ta-mw-onyesha = I will show her

That is why you see onyesha inside nitamwonyesha, not kuonyesha.