A tinta azul caiu num azulejo da cozinha, mas saiu com água quente.

Portuguese uses definite articles much more often than English. So a tinta azul is perfectly normal where English would simply say blue ink.

Here, a means the, but in many contexts it does not sound as specific or heavy as the does in English. It often just introduces the thing being talked about.

• a tinta = the ink
• a tinta azul = the blue ink

If you removed a, the sentence would sound less natural in this context.

In Portuguese, adjectives usually come after the noun.

So:

• tinta azul = blue ink
• literally: ink blue

This is the normal order for colors and many descriptive adjectives.

Because azul is one of those adjectives whose singular form is the same for masculine and feminine.

So you get:

• tinta azul = feminine singular
• carro azul = masculine singular

It only changes in the plural:

• tintas azuis
• carros azuis

So azul is not like branco / branca. It stays azul in the singular.

Num is the contraction of em + um.

• em = in / on / at
• um = a / one
• em um → num

So:

• num azulejo = on/in a tile depending on context, but here naturally on a tile

This kind of contraction is extremely common in Portuguese.

Other similar ones:

• na = em + a
• no = em + o
• nuns = em + uns
• numas = em + umas

Yes. Em um azulejo is grammatically correct.

However, num azulejo is much more natural in everyday Portuguese, especially in speech and normal writing.

So both mean the same thing, but:

• num azulejo = more natural/common
• em um azulejo = possible, slightly less compact

An azulejo is a ceramic tile, especially the kind commonly used on walls in Portugal and very strongly associated with Portuguese culture.

In this sentence, it just means a tile, most likely a kitchen wall tile.

So:

• azulejo = tile
• um azulejo da cozinha = a kitchen tile / a tile from the kitchen wall area

Because da cozinha means of the kitchen or more naturally in English, from the kitchen / in the kitchen area as a description of the tile.

• da = de + a
• de a cozinha → da cozinha

So um azulejo da cozinha means a tile of the kitchen, which English would usually express as a kitchen tile or a tile in the kitchen.

If you said num azulejo na cozinha, that would focus more on the location: on a tile in the kitchen.
The original sentence instead identifies which tile: a tile from the kitchen.

They are both in the pretérito perfeito: the simple past used for completed actions.

• cair → caiu = fell
• sair → saiu = came out / came off

Both are 3rd person singular, because the subject is singular:

• a tinta azul caiu...
• a tinta azul ... saiu...

This tense is used because the sentence describes two finished events:

1. the ink fell onto the tile
2. it came off with hot water

Yes, sair usually means to go out / to leave, but it also has a wider meaning: to come out, to come off, or to disappear from a surface.

With stains, marks, ink, and dirt, Portuguese often uses sair where English might use:

• come off
• wash out
• come out
• be removed

So here:

• A tinta ... saiu com água quente
  = The ink came off with hot water

This is a very common use.

Examples:

• A nódoa saiu. = The stain came out.
• Isto não sai. = This won’t come off.

Yes. The subject is still understood to be a tinta azul.

Portuguese often does not repeat the subject if it is already clear from the context.

So the full idea is:

• A tinta azul caiu num azulejo da cozinha, mas [a tinta azul] saiu com água quente.

In natural English, we might prefer to interpret it as the ink stain came off, but Portuguese simply keeps referring to the ink.

Because com água quente means using hot water in a general sense.

When Portuguese talks about the means or substance used, it often leaves out the article:

• com água quente = with hot water
• com sabão = with soap
• com álcool = with alcohol

If you said com a água quente, it would sound more like with the hot water — a specific hot water already identified in the conversation.

So:

• com água quente = general, natural here
• com a água quente = specific hot water

Because mas means but, and it introduces a contrast between two clauses:

• the blue ink fell onto a kitchen tile
• but it came off with hot water

In Portuguese, it is normal to put a comma before mas when it connects two full clauses like this.

Yes. Em is a very flexible preposition. Depending on the context, it can mean:

• in
• on
• at
• sometimes other related ideas

With cair em / num, Portuguese can express the idea of something falling onto a surface or into something. The exact English translation depends on the situation.

Here, because a tile is a surface, English naturally says:

• fell on a tile

So the Portuguese stays the same, but English chooses the most natural preposition.

A rough guide is:

• A tinta azul → uh TEEN-tuh uh-ZOOL
• caiu → kuh-YOO
• num azulejo → noom uh-zoo-LAY-zhoo
  in European Portuguese, unstressed vowels are often reduced, so it may sound more compressed
• da cozinha → duh koo-ZEE-nyuh
• mas saiu → mahz suh-YOO
• com água quente → kong AH-gwuh KEN-tuh

A few useful points for European Portuguese:

• unstressed a often sounds like a reduced vowel, not a clear English ah
• saiu and caiu both have the stressed ending -iu
• z in cozinha is voiced, like z in zero

If you want, I can also give you a more precise European Portuguese IPA transcription of the whole sentence.