Learn Hardcore Norwegian: Etter skiftet er jeg så trøtt at jeg bare vil sove. - After the shift I am so tired that I just want to sleep.

The finite verb (here: er) must come in second position in the sentence.
The first position can be the subject, or something else (like a time expression).

In your sentence, the time expression Etter skiftet is put first. That means the verb must come next, and the subject comes after the verb:

Normal order (no fronted element):
Jeg er så trøtt ... – I am so tired ...
With a time expression in front:
Etter skiftet er jeg så trøtt ... – After the shift am I so tired ... (literally)

So er jeg is required by the word-order rule when Etter skiftet is moved to the front.

Why is it etter skiftet and not etter skift or etter et skift?

The noun skift (shift, work shift) is neuter:

et skift – an/one shift
skiftet – the shift

In this sentence, you are talking about a specific, known shift (your work shift that day), so the definite form is natural:

etter skiftet = after the shift (the one we both know about)

Etter et skift (after a shift) is grammatically correct but sounds more general and less tied to a particular, known shift.

Etter skift without any article is not idiomatic here. Countable nouns in Norwegian normally need an article (indefinite or definite) in this kind of context.

What is the function of så ... at in så trøtt at jeg bare vil sove?

The pattern så + adjective + at means “so [adjective] that …” and introduces a result:

så trøtt at ... = so tired that ...

Structure in your sentence:

så trøtt – so tired
at jeg bare vil sove – that I just want to sleep

This construction is very common:

Det var så kaldt at vi gikk hjem.
It was so cold that we went home.

So at here is a conjunction meaning that, linking the degree of tiredness to its result.

Could you leave out at and say så trøtt jeg bare vil sove?

Not in this normal declarative sentence.

With the result clause, you need at:
… så trøtt at jeg bare vil sove.

If you remove at, you get something closer to an exclamation or poetic style, but then you also change the word order:

Så trøtt jeg (bare) vil sove! – How tired I just want to sleep! (unusual, stylistic, not a neutral statement)

For a neutral sentence explaining cause and result, you should keep at.

Why is it trøtt and not sliten? Aren’t they both “tired”?

Both trøtt and sliten can translate as tired, but they have slightly different nuances:

trøtt – sleepy-tired, low energy, you feel like sleeping.
sliten – worn-out, exhausted, used up (physically or mentally) from effort.

In your sentence:

så trøtt at jeg bare vil sove – emphasises sleepiness.
så sliten at jeg bare vil sove – would emphasise being worn out from work, but still entirely natural.

Both are possible. The choice depends on whether you want to stress sleepiness (trøtt) or being worn out (sliten).

Why is it jeg bare vil sove and not jeg vil bare sove? Does the position of bare change the meaning?

Normally, the most common word order is:

Jeg vil bare sove. – I just want to sleep.

Here bare clearly limits the verb phrase vil sove: you don’t want anything else.

Your sentence has:

jeg bare vil sove

This is also used in spoken Norwegian and is usually understood with the same meaning. It can sound a bit more informal or give a slight emphasis on bare as a comment on the whole situation:

I just want to sleep, that’s all / All I want is to sleep.

For learners, it is safest to use Jeg vil bare sove as the default pattern, but you will hear jeg bare vil sove in everyday speech.

What does bare mean in this sentence?

Bare most often means only / just.

In jeg bare vil sove, it means:

I only want to sleep
I just want to sleep

So it limits or narrows down what you want: sleeping is the only thing you feel like doing.

Is vil here “will” (future) or “want to”?

Here vil means want to, not a pure future tense:

jeg vil sove – I want to sleep

Norwegian vil has two main uses:

Volition / desire (like English want to):
- Jeg vil spise. – I want to eat.
Future-like meaning, but usually with some nuance (willingness, prediction, etc.):
- Det vil ta tid. – It will take time.

In jeg bare vil sove, it is clearly desire, so understand it as I just want to sleep.

Could you say jeg har bare lyst til å sove instead of jeg bare vil sove?

Yes, absolutely. Both are natural, with a small nuance difference:

Jeg vil sove. – I want to sleep. (short, direct)
Jeg har lyst til å sove. – I feel like sleeping / I would like to sleep.

Adding bare:

Jeg vil bare sove. – I just want to sleep.
Jeg har bare lyst til å sove. – I only feel like sleeping.

Har lyst til å can sound a bit softer or more polite, while vil is more direct. In everyday speech, both are very common.

What exactly does etter mean here, and how is it different from etterpå?

Etter is a preposition meaning after, and it must be followed by:

a noun phrase:
etter skiftet – after the shift
etter jobben – after work
or a clause introduced by “at”:
etter at skiftet er ferdig – after the shift is finished

Etterpå is an adverb meaning afterwards / later on, and it stands alone, without a noun after it:

Etterpå er jeg så trøtt. – Afterwards I am so tired.

You cannot say etterpå skiftet. That is ungrammatical.

Is skift always neuter? How do I know?

Yes, skift is a neuter noun:

et skift – an/one shift
flere skift – several shifts
skiftet – the shift
skiftene – the shifts

There is unfortunately no general rule that lets you always guess gender correctly; you mainly have to learn noun genders individually. Dictionaries usually mark gender:

skift n. (n = neuter) or skift (et).

Over time, you get a feel for common patterns, but in the beginning it’s safest to check a dictionary.

Could you say Etter jobben er jeg så trøtt at jeg bare vil sove instead? Is it the same?

Yes, that is also natural and very common:

Etter jobben er jeg så trøtt at jeg bare vil sove.
After work I’m so tired that I just want to sleep.

Nuance:

etter skiftet – focuses on one particular shift (useful if you work shifts, night shifts, etc.).
etter jobben – more generally after work.

Both are fine; choose based on what you want to emphasise.

Why is the verb er in the present tense, not a past tense like var?

Both tenses are possible, but they say different things:

Etter skiftet er jeg så trøtt ...
Suggests something that happens regularly or is generally true:
After (my) shift, I am (always) so tired that I just want to sleep.
Etter skiftet var jeg så trøtt ...
Refers to a specific past occasion:
After the shift (yesterday / that day), I was so tired that I just wanted to sleep.

So er here presents it as a habitual or typical situation.