Мій двоюрідний брат старший за мене на три роки.

Because брат is masculine singular nominative, and the possessive adjective мій has to agree with it.

Agreement here is:

• мій = masculine singular
• двоюрідний = masculine singular
• брат = masculine singular noun

Compare:

• мій брат = my brother
• моя сестра = my sister
• моє місто = my city
• мої друзі = my friends

So мій is the correct form because it matches брат.

It means male cousin.

Literally, двоюрідний is the usual adjective used in family terms for a cousin relationship, so:

• двоюрідний брат = male cousin
• двоюрідна сестра = female cousin

English uses one word, cousin, for both sexes, but Ukrainian usually specifies gender with брат or сестра.

Usually, standard everyday Ukrainian prefers the gender-specific forms:

• двоюрідний брат
• двоюрідна сестра

You may also hear borrowed words like:

• кузен = male cousin
• кузина = female cousin

But for a learner, двоюрідний брат / двоюрідна сестра is the safest and most standard choice.

Yes. Старший means older in this sentence.

It comes from старий = old, but старший is the comparative form, like older.

So:

• старий = old
• старший = older

With people, старший is the normal way to say older when comparing ages:

• Він старший за мене. = He is older than me.
• Моя сестра старша за мене. = My sister is older than me.

Notice that старший changes for gender:

• masculine: старший
• feminine: старша
• neuter: старше
• plural: старші

In Ukrainian, a very common way to make comparisons is:

• comparative + за + accusative

So:

• старший за мене = older than me

That is why you see за мене.

Another common pattern is:

• старший від мене = older than me

Both are natural.

You can also sometimes use ніж, but старший за мене is one of the most standard and straightforward ways to say it.

Because after за in this comparison pattern, Ukrainian uses the accusative case.

The pronoun я changes like this:

• nominative: я = I
• accusative: мене = me

So:

• я = I
• за мене = than me

That is why я would be wrong here.

It means by three years.

So the full structure is:

• старший за мене = older than me
• на три роки = by three years

Together:

• старший за мене на три роки = older than me by three years

This phrase expresses the amount of difference.

You can use the same pattern in many other sentences:

• Вона молодша за мене на два роки. = She is two years younger than me.
• Він вищий за мене на десять сантиметрів. = He is taller than me by ten centimeters.

Because after the numeral три, Ukrainian uses the form роки.

A helpful basic rule is:

• 1 → рік
• 2, 3, 4 → роки
• 5 and higher → років

So:

• один рік = one year
• два роки = two years
• три роки = three years
• чотири роки = four years
• п’ять років = five years

That is why на три роки is correct.

It can change somewhat. Ukrainian word order is more flexible than English word order.

The sentence:

• Мій двоюрідний брат старший за мене на три роки.

is completely natural.

But you may also hear variations such as:

• Мій двоюрідний брат на три роки старший за мене.

This means the same thing. The difference is mostly one of emphasis and rhythm.

For learners, the original sentence is a very good model to follow.

Yes, often you can.

For example:

• Двоюрідний брат старший за мене на три роки.

This can work if the listener already knows whose cousin you mean.

However, мій makes the sentence clearer and more complete, especially when you are introducing the person for the first time.

So:

• мій двоюрідний брат = my cousin
• двоюрідний брат = cousin / a cousin, depending on context

This sentence is natural and idiomatic.

Мій двоюрідний брат старший за мене на три роки. is a perfectly normal way to say this.

A very common alternative is:

• Мій двоюрідний брат на три роки старший за мене.

Both are good. If you are learning, either version is worth remembering.