Můj bratr mi ukazuje starou fotku z minulého roku u moře.

Because můj bratr is the subject of the sentence, so it is in the nominative singular.

• bratr = brother
• můj must agree with bratr
• bratr is masculine singular, so the correct nominative form is můj

So:

• můj bratr = my brother
• compare:
  • moje sestra = my sister
  • moje auto = my car

Here, můj is simply the normal nominative masculine singular form.

Mi is the dative form of the pronoun já and means to me / for me.

In this sentence, the brother is showing the photo to me, so Czech uses the dative:

• můj bratr mi ukazuje... = my brother is showing me...

A very useful way to think about it:

• mi / mně = to me
• mě / mne = me

So here you need mi because the idea is showing something to someone.

They both mean to me, but they are used differently.

• mi is the short, unstressed form
• mně is the full, stressed form

In a normal neutral sentence, mi is much more natural:

• Můj bratr mi ukazuje... = neutral and natural

You use mně more often:

• for emphasis
• after prepositions
• when the short clitic form cannot be used

So in this sentence, mi is the expected choice.

Ukazuje is the 3rd person singular present tense form of ukazovat.

That matches the subject můj bratr:

• já ukazuju / ukazuji = I show / am showing
• ty ukazuješ = you show
• on ukazuje = he shows / is showing

So:

• můj bratr ukazuje = my brother shows / is showing

Czech present tense often covers both English shows and is showing, depending on context.

Also, ukazovat is imperfective, so it fits an ongoing action well.

Because starou fotku is the direct object of the verb, so it has to be in the accusative.

The basic form is:

• stará fotka = an old photo

But after a verb like ukazuje when the photo is the thing being shown, Czech uses the accusative:

• stará → starou
• fotka → fotku

So:

• ukazuje starou fotku = is showing an old photo

This is a very common pattern in Czech:

• nominative: stará fotka
• accusative: starou fotku

Yes. Fotku is the accusative singular of fotka, which is a very common everyday word for photo.

So:

• fotka = photo, picture
• fotku = photo, picture as a direct object

You could also say fotografii, from fotografie:

• ukazuje mi starou fotografii = more formal or bookish
• ukazuje mi starou fotku = normal everyday Czech

So fotku is not wrong or childish; it is just the usual spoken-style word.

Because the preposition z requires the genitive case.

The basic noun is:

• rok = year

After z, it changes to genitive:

• z roku = from the year / of the year

The adjective must match:

• minulý rok = last year
• z minulého roku = from last year / of last year

So both words are in the genitive:

• minulého = genitive singular
• roku = genitive singular

This is a very common Czech pattern:

• z domu = from the house
• z města = from the city
• z minulého roku = from last year

U moře means by the sea, at the seaside, or sometimes more loosely at the sea.

The preposition u usually means by, near, at and it takes the genitive case.

With moře:

• nominative: moře
• genitive: moře

The form happens to stay the same, so you do not see a visible change here.

A useful comparison:

• u moře = by the sea / at the seaside
• v moři = in the sea
• k moři = to the sea

So u moře does not mean in the sea.

Because mi is a short unstressed pronoun, often called a clitic, and Czech clitics usually go in second position in the sentence.

Here the first unit is Můj bratr, and then the clitic mi comes right after it:

• Můj bratr mi ukazuje...

That is very natural Czech word order.

English speakers often want to place it later because English does not work this way, but in Czech:

• Můj bratr mi ukazuje... = natural
• Můj bratr ukazuje mi... = unnatural in a neutral sentence

So this is not random word order; it follows a common Czech pattern.

In this sentence, u moře most naturally goes with the photo or the situation shown in the photo.

So the most likely interpretation is:

• my brother is showing me an old photo from last year, taken by the sea / from a seaside holiday

Czech can sometimes leave this slightly open, but with this word order, many people will connect u moře with fotku z minulého roku.

If you wanted to make it clearer that the brother is by the sea while showing it, you would usually place u moře earlier or restructure the sentence.

Yes, it is grammatically natural in Czech.

The only small issue is meaning: starou fotku z minulého roku can sound slightly context-dependent, because a photo from last year is not very old in an absolute sense.

But starou can still make sense if you mean:

• an older photo, as opposed to a newer one
• a photo that has been around for a while
• a photo from an earlier moment

If you only wanted to say a photo from last year, you might simply say:

• Můj bratr mi ukazuje fotku z minulého roku.

So starou is fine, but it adds an extra nuance rather than being strictly necessary.