Ég lét klippa hárið í gær, því ég hafði ekki tíma til að gera það sjálf.

Lét is the past tense of láta. In this sentence, láta + infinitive is a very common Icelandic way to express having something done or getting someone to do something.

So:

• Ég lét klippa hárið = I had my hair cut
• literally, something like I caused the hair to be cut

This is similar to English have/get something done:

• Ég lét laga bílinn = I had the car repaired
• Hún lét þrífa húsið = She had the house cleaned

So the sentence does not mean that the speaker personally did the cutting.

After láta, Icelandic normally uses the bare infinitive, without að.

So you say:

• Ég lét klippa hárið
• not Ég lét að klippa hárið

This is similar to how some verbs in English take a bare infinitive:

• I made him go
• not I made him to go

Many Icelandic verb combinations work differently, so this is something you mainly have to learn verb by verb. With láta, the bare infinitive is standard.

Icelandic often uses the definite noun for body parts when it is already obvious whose body part is meant.

So:

• Ég lét klippa hárið naturally means I had my hair cut

Even though the word my is not stated, it is understood from context. This is very common in Icelandic.

Compare:

• Hún þvoði hendurnar = She washed her hands
• literally: She washed the hands

Using hár mitt is possible in some contexts, but here it would sound less natural than hárið.

Hárið is in the accusative singular definite form of hár.

That is because it is the direct object of klippa:

• klippa hárið = cut the hair

Even though the whole phrase is governed by lét, the noun is still the object of the infinitive klippa.

So the structure is roughly:

• Ég = subject
• lét = past tense verb
• klippa hárið = infinitive phrase, where hárið is the object of klippa

In this sentence, því is a conjunction meaning because.

So:

• ..., því ég hafði ekki tíma ... = ..., because I didn’t have time ...

This is a very common written and spoken way to introduce a reason.

You may also see other ways to say because, such as:

• af því að
• vegna þess að

These are often a bit fuller or more explicit. In this sentence, því is completely natural and concise.

The pattern is:

• hafa tíma til að + infinitive

It means to have time to do something.

So:

• ég hafði ekki tíma til að gera það = I did not have time to do it

Breakdown:

• hafði = had
• ekki tíma = no time / not time
• til að gera = to do

This is a fixed and very common Icelandic structure.

Examples:

• Ég hef tíma til að lesa. = I have time to read.
• Við höfðum ekki tíma til að borða. = We didn’t have time to eat.

Það refers back to the action of cutting the hair.

So:

• að gera það sjálf = to do it myself

Here það stands for something like:

• to cut my own hair
• or more generally to take care of that myself

Icelandic often uses það like English it to refer back to an action or situation that has already been mentioned.

Sjálf agrees with the gender of the speaker.

Since the sentence uses sjálf, the speaker is female.

So:

• ég ... sjálf = I ... myself said by a woman
• ég ... sjálfur = I ... myself said by a man

This is an important point in Icelandic: words like sjálfur/sjálf/sjálft change to match gender, number, and case.

Here the form is feminine because the speaker is feminine.

Yes, in this sentence it works very much like English myself.

• gera það sjálf = do it myself

It emphasizes that the speaker would have done the hair-cutting personally, rather than having someone else do it.

In Icelandic, sjálfur/sjálf/sjálft is often used for this kind of emphasis:

• Ég gerði það sjálf. = I did it myself.
• Hann sá það sjálfur. = He saw it himself.

Because því introduces a subordinate clause, and subordinate clauses in Icelandic do not use the main-clause inversion pattern.

So you get:

• því ég hafði ekki tíma ...

not:

• því hafði ég ekki tíma ...

Compare:

Main clause:

• Í gær lét ég klippa hárið. Here inversion happens because Í gær comes first.

Subordinate clause:

• ..., því ég hafði ekki tíma ... No inversion after því.

This is a very common thing learners notice in Icelandic word order.

Yes, it is a normal and idiomatic way to say it.

It implies that someone else cut the speaker’s hair, usually in a practical situation such as a hairdresser’s salon, though the sentence itself does not explicitly say where or by whom.

This construction is very useful in Icelandic for services or actions done by others:

• Ég lét mála húsið. = I had the house painted.
• Við létum laga tölvuna. = We had the computer repaired.

So this is exactly the kind of structure learners should get comfortable with.

Í gær means yesterday, and here it comes after hárið:

• Ég lét klippa hárið í gær

That placement is completely natural. Icelandic is fairly flexible with adverb placement, especially with time expressions.

You could also say:

• Í gær lét ég klippa hárið

That version puts more emphasis on yesterday and also triggers the usual main-clause inversion:

• Í gær lét ég ... not Í gær ég lét ...

So the original word order is neutral and natural, while fronting í gær gives it more focus.

It only says that the speaker did not have time.

The reason clause is:

• því ég hafði ekki tíma til að gera það sjálf = because I didn’t have time to do it myself

So the sentence does not imply lack of ability. It only gives lack of time as the reason for having someone else do it.