Bróðir minn veit ekki hvaða bindi hann á að setja á sig.

In ordinary Icelandic, possessive pronouns like minn often come after the noun:

• bróðir minn = my brother

This is the most neutral, everyday way to say it.
Minn bróðir is possible, but it sounds more marked, emphatic, or literary.

Also, bróðir is in the nominative singular because it is the subject of the sentence.

Bróðir is nominative singular masculine.

It is nominative because it is the subject of veit ekki (does not know):

• Bróðir minn = the person doing the knowing

The noun bróðir is an irregular masculine noun, so its forms are worth learning separately.

Hvaða means which or what kind of, depending on context.

Here it introduces an embedded question:

• veit ekki hvaða bindi... = doesn’t know which tie...

A useful point: hvaða is generally treated as indeclinable in modern Icelandic, so it does not change form for gender, number, or case.

Because it is an indirect question embedded inside a larger sentence.

The whole sentence is:

• Bróðir minn veit ekki ... = My brother does not know ...

What he does not know is:

• hvaða bindi hann á að setja á sig = which tie he is supposed to put on

Compare:

• Direct question: Hvaða bindi á hann að setja á sig?
  Which tie is he supposed to put on?
• Indirect question: ... veit ekki hvaða bindi hann á að setja á sig.
  ... doesn’t know which tie he is supposed to put on.

Because this is a subordinate clause (an embedded clause), not a main-clause question.

In an indirect question, Icelandic usually uses more statement-like word order:

• hvaða bindi hann á að setja á sig

If it were a direct question, you would normally get inversion:

• Hvaða bindi á hann að setja á sig?

So the word order helps show whether the clause is direct or embedded.

Here bindi means tie or necktie.

It is a neuter noun. In this sentence it is the direct object of setja á sig.

One small thing that can confuse learners: bindi can have other meanings in other contexts, but in clothing contexts it commonly means tie.

It is in the accusative singular, because it is the direct object of setja.

However, with many neuter nouns, the nominative and accusative singular look the same, so you just see:

• bindi

So even though the form is unchanged, its grammatical role here is accusative.

Here á að + infinitive is not about possession. It is the common Icelandic expression:

• eiga að + infinitive = to be supposed to, ought to, should

So:

• hann á að setja á sig = he is supposed to put on

This is a very common construction in Icelandic.
It does not mean that he owns something here.

Here að is the infinitive marker, similar to English to before a verb.

So:

• setja = put
• að setja = to put

In the full construction:

• á að setja = is supposed to put / is to put

So að belongs with the infinitive setja.

Because the action refers back to the subject hann (he), Icelandic uses the reflexive pronoun sig in the 3rd person.

So:

• hann ... setja á sig = he ... put on himself

If you said á hann, it would normally mean putting it on him as an object, not reflexively in the normal put on oneself sense.

This is a very important pattern:

• ég set á mig = I put on
• þú setur á þig = you put on
• hann/hún setur á sig = he/she puts on

Only the 3rd person uses sig.

Setja á sig literally means something like put onto oneself, and it is a normal way to say put on for clothes or accessories.

So:

• setja bindi á sig = put a tie on
• setja hatt á sig = put a hat on

The preposition á is part of this expression.
A learner should understand it as a unit:

• setja á sig = put on oneself

Because the phrase is indefinite: which tie, not which the tie.

In Icelandic, hvaða already works as the determiner here, so you do not normally add the definite article:

• hvaða bindi = which tie

Using a definite form such as bindið would change the meaning and would not fit the normal structure here.

That is the normal position in a main clause:

• veit ekki = does not know

Icelandic often places ekki after the finite verb in simple main-clause statements like this.

So:

• Bróðir minn veit ekki... = My brother does not know...

This is the most natural word order here.