Poslije kiše na nebu je bila duga.

Because poslije normally takes the genitive case.

The base noun is kiša = rain.
Its genitive singular form is kiše.

So:

• kiša = rain
• poslije kiše = after the rain / after rain

This is a very common pattern in Croatian:

• poslije škole = after school
• poslije ručka = after lunch
• poslije rata = after the war

Here na nebu is the normal Croatian way to say in the sky / in the heavens.

Even though na often means on, Croatian does not always match English prepositions directly. With nebo (sky), the idiomatic expression is:

• na nebu = in the sky

So this is something learners usually just have to memorize as a set phrase.

Because after na in a location meaning, Croatian uses the locative case.

The noun is:

• nebo = sky

Its locative singular is:

• nebu

So:

• na nebu = in the sky

This is the same idea as:

• u gradu = in the city
• na stolu = on the table

That is how the past tense is formed in Croatian.

Croatian past tense usually uses:

1. the present tense of biti (to be)
2. a past participle

So here:

• je = is / has, the 3rd person singular auxiliary
• bila = the past participle, feminine singular

Together:

• je bila = was

So na nebu je bila duga means there was a rainbow in the sky / a rainbow was in the sky.

Because duga is a feminine singular noun, and the past participle must agree with the subject in gender and number.

So:

• masculine singular → bio
• feminine singular → bila
• neuter singular → bilo

Since duga is feminine, the sentence uses bila.

Yes. Duga can be:

1. a noun meaning rainbow
2. a feminine form of the adjective dug = long

In this sentence, it is clearly the noun rainbow.

So:

• duga cesta = a long road
• na nebu je bila duga = there was a rainbow in the sky

Croatian often relies on context to show which meaning is intended.

Because Croatian does not have articles like English a/an/the.

So duga can mean:

• a rainbow
• the rainbow

The exact meaning depends on context.

That is very normal in Croatian. Learners coming from English often expect an article, but Croatian simply does not use one.

Yes, Croatian word order is more flexible than English word order.

This sentence starts with the time phrase and then the place phrase:

• Poslije kiše = after the rain
• na nebu = in the sky
• je bila duga = there was a rainbow

This order sounds natural and puts the scene first.

Other orders are also possible, for example:

• Duga je bila na nebu poslije kiše.
• Na nebu je bila duga poslije kiše.

These are grammatically possible, but they may sound a little different in emphasis. The original sentence nicely sets the scene first.

Because je is a clitic in Croatian. Clitics usually go in the second position of the clause, not wherever an English speaker might expect them.

In na nebu je bila duga:

• na nebu acts as the first unit
• je comes right after that unit

This is very typical Croatian syntax.

So even though je bila functions together as the past tense, the little word je often appears early in the sentence because of clitic placement rules.

Yes. Nakon kiše is also correct and means the same thing: after the rain.

So both are possible:

• Poslije kiše na nebu je bila duga.
• Nakon kiše na nebu je bila duga.

In many contexts they are interchangeable. Poslije is very common in everyday speech, while nakon can sound a little more formal or neutral, depending on context.

Yes, it is natural and correct. But Croatian also has other natural ways to express the same idea.

For example:

• Poslije kiše na nebu se pojavila duga. = After the rain, a rainbow appeared in the sky.
• Poslije kiše pojavila se duga na nebu.

Your sentence is a simple descriptive one: it states that the rainbow was there. Other versions may sound a bit more dynamic because they focus on the rainbow appearing.