Hún á eftir að hengja gardínurnar upp í stofunni í kvöld.

Here á eftir að + infinitive is an idiomatic expression. It usually means something like:

• still has to
• is yet to
• will later

So Hún á eftir að hengja gardínurnar upp means that this action has not happened yet, but she is expected to do it later.

It is not understood literally as owns after to. You should learn eiga eftir að as a set expression.

Á is the 3rd person singular present form of eiga.

The pattern is:

• ég á eftir að
• þú átt eftir að
• hann / hún / það á eftir að
• við eigum eftir að

Because the subject is hún (she), the verb has to be á.

So the structure is:

Hún + á + eftir að + infinitive

It often has a future meaning, yes, but the nuance is a little more specific than a simple future.

Hún á eftir að hengja gardínurnar upp í kvöld does not just mean she will hang the curtains tonight. It suggests that this is something still remaining to be done.

So the feeling is closer to:

• She still has to hang the curtains tonight
• She is yet to hang the curtains tonight

Icelandic often uses present-tense forms and expressions like this instead of a separate future tense.

Because after eiga eftir, Icelandic uses the infinitive with að:

• eiga eftir að gera
• eiga eftir að fara
• eiga eftir að hengja

So að hengja is simply the infinitive construction required by this expression.

This is something you mostly just learn with the pattern:

eiga eftir að + verb

The basic verb hengja means to hang. With upp, it becomes hengja upp, which means hang up or put up.

So:

• hengja gardínurnar = hang the curtains
• hengja gardínurnar upp = hang the curtains up / put the curtains up

The upp is a verb particle, similar to particles in English phrasal verbs such as put up or hang up. In Icelandic, that particle is often written as a separate word.

Yes, that is possible.

Both of these are natural:

• hengja gardínurnar upp
• hengja upp gardínurnar

The version in your sentence places the object before the particle:

• hengja gardínurnar upp

That is a very common pattern, especially with a full noun phrase like gardínurnar.

So the sentence is perfectly normal as written, but the other order can also occur.

Gardínurnar means the curtains.

It comes from:

• singular: gardína = curtain
• plural: gardínur = curtains
• definite plural: gardínurnar = the curtains

The definite article in Icelandic is usually attached to the noun as an ending, instead of being a separate word like English the.

In this sentence, gardínurnar is the direct object of hengja. Syntactically that means accusative, although for this noun the nominative and accusative plural look the same.

Because í can take different cases depending on meaning:

• accusative for motion into
• dative for location in

Here the phrase means in the living room, describing where the action happens, so Icelandic uses the dative:

• í stofunni = in the living room

Compare:

• Hún fer í stofuna = She goes into the living room
• Hún er í stofunni = She is in the living room

In your sentence, the location is the living room, so stofunni is correct.

Í kvöld is a fixed time expression meaning tonight.

Icelandic often uses short set expressions for times:

• í dag = today
• í morgun = this morning
• í kvöld = tonight
• í nótt = tonight / during the night

These normally do not take the definite article in ordinary use. So í kvöld is the normal expression here.

That is a normal Icelandic word order.

After the main verb phrase, Icelandic often places adverbial phrases such as:

• place: í stofunni
• time: í kvöld

So this order is very natural:

Hún á eftir að hengja gardínurnar upp í stofunni í kvöld.

You can move elements around for emphasis, but the sentence as written is neutral and idiomatic. For example, í kvöld could be moved earlier if you wanted to stress the time, but it does not need to be.

Yes. That is one reason eiga eftir að can feel a little tricky to English speakers.

Depending on context, it can suggest:

• an action is still unfinished
• an action is going to happen later
• sometimes even that something is bound to happen

In this sentence, the most natural understanding is that hanging the curtains is a task that remains to be done, and the planned time is tonight. So both ideas are present:

• it has not happened yet
• it will happen later tonight