Eu gosto de kiwi ao pequeno-almoço.

Because gostar normally takes de in Portuguese.

So:

• gostar de + noun = to like something
• gostar de + infinitive = to like doing something

Examples:

• Gosto de kiwi. = I like kiwi.
• Gosto de ler. = I like reading.

So Eu gosto de kiwi is the correct structure.
Eu gosto kiwi would be wrong.

You can often leave out Eu.

Portuguese usually drops subject pronouns when the verb already makes the subject clear.

So both are natural:

• Eu gosto de kiwi ao pequeno-almoço.
• Gosto de kiwi ao pequeno-almoço.

The version with Eu may sound a little more explicit or emphatic, but it is not necessary in most contexts.

Here kiwi is being used in a general sense, like a food you like eating, not one specific kiwi.

So:

• gosto de kiwi = I like kiwi / I like eating kiwi
• gosto de um kiwi = I like one kiwi / a particular kiwi
• gosto de kiwis = I like kiwis

In Portuguese, using the singular without an article is very common when talking about foods in a general way.

Normally, kiwi is masculine in European Portuguese when you use an article:

• o kiwi
• um kiwi

Examples:

• O kiwi é doce. = Kiwi is sweet.
• Comi um kiwi. = I ate a kiwi.

But in your sentence, there is no article, so you do not see the gender directly:

• gosto de kiwi

Ao pequeno-almoço means for breakfast or at breakfast.

Ao is a contraction of:

• a + o = ao

Here, o pequeno-almoço is a masculine noun phrase, and Portuguese often uses this pattern with meals and times in certain expressions.

So:

• ao pequeno-almoço = at breakfast / for breakfast

This is a very natural European Portuguese way to express it.

Because pequeno-almoço is a fixed compound noun in European Portuguese.

It literally comes from:

• pequeno = small
• almoço = lunch

Historically, the idea is something like a small meal before the main meal.

In modern Portuguese from Portugal, pequeno-almoço simply means breakfast, and it is normally written with a hyphen.

Yes. Pequeno-almoço is the normal term in Portugal.

In Brazilian Portuguese, the usual expression is:

• café da manhã

So a Brazilian would usually say something like:

• Eu gosto de kiwi no café da manhã.

But in Portugal, ao pequeno-almoço is the natural choice.

Yes, absolutely.

Both are correct, but they feel slightly different:

• Gosto de kiwi ao pequeno-almoço. = I like kiwi for breakfast.
• Gosto de comer kiwi ao pequeno-almoço. = I like eating kiwi for breakfast.

The version without comer is shorter and very natural.
The version with comer is a bit more explicit.

A simple approximate pronunciation is:

eh-oo GOSH-too d(ə) KEE-wee ow pə-KEH-noo ahl-MO-soo

A few notes:

• gosto: the s sounds like sh in European Portuguese
• de is often reduced, almost like d(ə)
• ao sounds roughly like ow
• pequeno-almoço has the main stress on -moço

This is only an approximation for English speakers, but it helps as a starting point.

Yes. Portuguese word order is flexible.

These are all possible:

• Eu gosto de kiwi ao pequeno-almoço.
• Ao pequeno-almoço, eu gosto de kiwi.
• Gosto de kiwi ao pequeno-almoço.

The original version is the most neutral and natural if you are simply stating a preference.

Yes, but it means something different.

• Eu gosto de kiwi ao pequeno-almoço. = I like kiwi for breakfast.
  This talks about preference.
• Eu como kiwi ao pequeno-almoço. = I eat kiwi for breakfast.
  This talks about habit or fact.

So both are correct, but they are not interchangeable in meaning.

There actually is an article there — it is just merged into ao.

• a + o pequeno-almoço becomes ao pequeno-almoço

So ao already includes the masculine singular definite article o.

That is why you do not write ao o pequeno-almoço.