Idan babu kankara a firji, zan sha ruwa kawai.

Idan introduces a condition, so here it means if.

• Idan babu kankara a firji = If there is no ice in the fridge...

In some contexts, idan can also be understood more like when, especially in general truths, but in this sentence if is the best match.

Because babu is the normal Hausa way to say that something is absent or does not exist.

• babu kankara = there is no ice
• babu mutane = there are no people

By contrast, ba ... ba usually negates a verb or a predicate:

• Ba na sha ba = I do not drink
• Ba sanyi ba ne = It is not cold

So in your sentence, babu is the right choice because the idea is there is no ice, not ice does not do something.

Yes: akwai.

• Akwai kankara a firji. = There is ice in the fridge.
• Babu kankara a firji. = There is no ice in the fridge.

These two words are very useful as a pair:

• akwai = there is / there are
• babu = there is no / there are no

A very literal breakdown is:

• babu = there is not / there does not exist
• kankara = ice
• a firji = in the fridge

So the whole phrase is basically:

there-is-no ice in the fridge

English uses there is/there are, but Hausa handles this kind of idea with words like akwai and babu.

Here a is a locative preposition, meaning in.

So:

• a firji = in the fridge

This same a can also mean at, on, or sometimes to, depending on context. In this sentence, in is the natural translation.

It is a loanword, ultimately from English fridge.

That is very common in Hausa, especially for modern items and technology. So firji is perfectly normal everyday Hausa, even though its origin is foreign.

Zan means I will or I am going to.

So:

• zan sha = I will drink

You can think of zan as the future marker together with the I subject. In ordinary learning terms, it is easiest to remember it as one unit meaning I will.

Because that is the normal Hausa pattern.

In this kind of sentence, the subject-plus-tense marker comes before the main verb:

• zan sha = I will drink
• zai zo = he will come
• za mu tafi / zamu tafi = we will go

So zan sha is the expected word order.

In this sentence, yes: sha ruwa clearly means drink water.

More broadly, sha can be used in a few related senses involving consuming or taking something in, but for a learner, the most important meaning here is simply:

• sha = drink

So zan sha ruwa = I will drink water.

Kawai means only or just, and it usually comes after the word or phrase it is limiting.

So:

• ruwa kawai = only water / just water

That is why the sentence ends with kawai. It tells you that water is the only thing the speaker will drink in that situation.

Because Hausa does not use articles the same way English does.

English often needs words like a, an, and the. Hausa usually does not. The listener understands whether something is general or specific from context.

So:

• kankara can mean ice or the ice, depending on context
• ruwa can mean water or the water, depending on context

That is normal Hausa grammar, not something missing.

Yes. You could also say:

Zan sha ruwa kawai idan babu kankara a firji.

That still means the same thing.

The difference is mostly about emphasis and flow:

• Idan babu kankara a firji, zan sha ruwa kawai.
  Starts with the condition.
• Zan sha ruwa kawai idan babu kankara a firji.
  Starts with the main action.

Both are natural.

It is normal, natural everyday Hausa.

Nothing in it is unusually formal or literary:

• idan = common
• babu = very common
• firji = everyday word
• zan sha ruwa kawai = very natural speech

So this is a good practical sentence pattern to learn.