Mér er illt í bakinu eftir langan göngutúr.

Because Icelandic often expresses physical sensations with a dative experiencer rather than a nominative subject.

• ég = I (nominative)
• mér = to me / for me (dative)

So Mér er illt is literally something like It is painful for me or I feel pain, not a direct I hurt structure.

This same pattern appears in very common sentences like:

• Mér er kalt = I am cold
• Mér er heitt = I am hot
• Mér er illt = I am in pain / it hurts

Because this is a very common Icelandic way to describe a bodily state: dative pronoun + vera + adjective.

So instead of building the sentence around a verb like English hurt, Icelandic often says something closer to:

• to me is painful
• to me is cold
• to me is bad

In this sentence:

• Mér = to me / for me
• er = is
• illt = painful / bad

That whole chunk, Mér er illt, functions like I’m hurting / I’m in pain.

Because this is an impersonal construction, and in this kind of pattern Icelandic usually uses the adjective in the neuter singular default form.

The adjective is illur / ill / illt:

• masculine: illur
• feminine: ill
• neuter: illt

Here it does not agree with mér, because mér is not a nominative subject. So Icelandic uses the default neuter singular: illt.

This is the same reason you get:

• Mér er kalt
• Mér er heitt
• Mér er illt

Í bakinu means in the back or more naturally in my back in this sentence.

The noun is:

• bak = back

After í, Icelandic uses:

• accusative for motion into
• dative for location in

Here the pain is located in the back, not moving into it, so í takes the dative.

That gives:

• bak → bakinu = the back in the dative singular

So:

• í bakinu = in the back

Because Icelandic often uses the definite article with body parts when it is already obvious whose body part is meant.

So where English often says:

• my back
• my head
• my hand

Icelandic very often says:

• bakið / bakinu = the back
• höfuðið / höfðinu = the head
• höndin / hendinni = the hand

In a sentence like Mér er illt í bakinu, it is already clear that the speaker means their own back, so a possessive like mitt bak would usually sound unnecessary.

It is accusative singular masculine.

• göngutúr is a masculine noun
• langan is the accusative singular masculine form of langur

So:

• langur göngutúr = a long walk
• langan göngutúr = a long walk (accusative)

In this sentence, that happens because eftir is taking the accusative in this use, so the adjective has to match.

A useful thing to notice: the noun göngutúr itself looks the same in more than one case, so the adjective langan is what clearly shows you the case here.

It is a compound noun:

• ganga = to walk
• túr = trip / round / excursion

So göngutúr is literally something like a walking trip, but in natural English it usually means:

• walk
• stroll
• hike
  depending on context

In this sentence, langan göngutúr is simply a long walk.

Yes. Icelandic word order is fairly flexible, especially when you want to emphasize one part of the sentence.

The version you have is very natural:

• Mér er illt í bakinu eftir langan göngutúr.

But you could also move the time/cause phrase forward for emphasis:

• Eftir langan göngutúr er mér illt í bakinu.

That still means the same thing, but it puts more focus on after a long walk.

So the end position of eftir langan göngutúr is just the most neutral, everyday ordering here.