Bango linaloonyesha ratiba ya mtihani limebandikwa ukutani.

Because bango belongs to noun class 5 in Swahili, and words that agree with it often take the class 5 agreement prefix li-.

So in this sentence:

• bango = poster / notice
• li-na-lo-onyesha = it-that-is-showing
• li-me-bandikwa = it-has-been-posted/stuck

That li- is the agreement marker matching bango.

linaloonyesha can be broken down like this:

• li- = subject marker agreeing with bango (class 5)
• -na- = present / ongoing / general descriptive tense
• -lo- = relative marker for class 5, meaning something like that/which
• onyesha = show, display

So linaloonyesha literally means something like:

that is showing / which shows

In natural English here, it works like:

the poster that shows the exam schedule
or
the poster showing the exam schedule

Because Swahili often builds relative clauses directly into the verb instead of using a separate word like English that or which.

So instead of saying something structurally like:

bango that shows...

Swahili commonly says:

bango linaloonyesha...

The relative meaning is built into the verb form itself.

Yes, you could. That version is also grammatical.

• bango linaloonyesha... = a more compact relative construction
• bango ambalo linaonyesha... = a version using ambalo for which/that

Both mean basically the same thing.

The version in your sentence, linaloonyesha, is very natural and common.

This is a noun phrase meaning:

the schedule of the exam
or more naturally in English, the exam schedule

Breakdown:

• ratiba = schedule, timetable
• ya = of
• mtihani = exam

So ya links the two nouns, like of in English.

Here, ya agrees with ratiba, not with mtihani.
That is because ratiba is the head noun of the phrase.

Because the connector must agree with the noun that comes before it, which here is ratiba.

ratiba is in the N class (often called class 9/10), and one of its associative markers is ya.

So:

• ratiba ya mtihani = exam schedule
• literally: schedule of exam

This agreement pattern is very common in Swahili noun phrases.

limebandikwa can be broken down as:

• li- = subject marker agreeing with bango
• -me- = perfect marker, often has/have
• bandik- = verb root meaning stick, paste, post
• -w- = passive suffix
• -a = final vowel

So the basic idea is:

it has been posted / it has been stuck up

Because of the passive -w-, the poster is receiving the action, not doing it.

The -me- usually gives a perfect sense: a completed action with present relevance.

So limebandikwa suggests:

has been posted
or
has been put up

That fits the context well, because the poster is now up on the wall.

Compare roughly:

• limebandikwa = has been posted / is up now
• libandikwa = was posted
• linabandikwa = is being posted

The passive is used because the sentence is focused on the poster, not on the person who put it there.

So:

• limebandikwa ukutani = has been posted on the wall
• the doer is not mentioned

If you wanted to mention the doer, you could add it:

• Bango ... limebandikwa ukutani na mwalimu.
  = The poster ... has been posted on the wall by the teacher.

Swahili uses the passive very naturally in this kind of sentence.

ukutani means on the wall or at the wall.

It comes from:

• ukuta = wall
• -ni = locative suffix

So ukutani is a locative form meaning something like at/on the wall.

This -ni ending is very common in Swahili for places and locations.

Examples:

• nyumbani = at home
• shuleni = at school
• mezani = on the table
• ukutani = on the wall

They are very close in meaning, and in many contexts either one works.

• ukutani = on/at the wall
• kwenye ukuta = on the wall / on the surface of the wall

In this sentence, ukutani is short, natural, and perfectly idiomatic.

Because Swahili does not normally have articles like English the or a/an.

So bango can mean:

• a poster
• the poster

The exact meaning depends on context.

In this sentence, context usually makes it sound definite in English:

The poster showing the exam schedule has been posted on the wall.

The structure is:

Bango + relative clause + main verb + location

So:

• Bango = the poster
• linaloonyesha ratiba ya mtihani = that shows the exam schedule
• limebandikwa = has been posted
• ukutani = on the wall

That gives:

Bango linaloonyesha ratiba ya mtihani limebandikwa ukutani.

This word order is very normal in Swahili.