Puer fulgur procul supra montes miratur.

Because puer is in the nominative singular, the case normally used for the subject.

A few clues work together:

• puer = nominative singular of puer, pueri = boy
• miratur = he/she/it marvels at, admires, so the verb is 3rd person singular
• fulgur could be nominative or accusative, but puer cannot be accusative

So the grammar points to:

• puer = subject
• fulgur = direct object

In other words, the structure is The boy ... admires the lightning rather than The lightning admires the boy.

Here fulgur is accusative singular, the direct object of miratur.

It looks the same as the dictionary form because fulgur is a neuter third-declension noun:

• nominative singular: fulgur
• accusative singular: fulgur

That is very normal for neuter nouns in Latin:
nominative and accusative are often identical.

Its dictionary entry is:

• fulgur, fulguris = lightning, flash of lightning

Because miror, mirari is a deponent verb.

Deponent verbs:

• have passive forms
• but active meanings

So miratur looks like a passive form because of the ending -tur, but it actually means:

• he marvels at
• he admires
• he wonders at

This is one of the first important Latin patterns learners have to get used to.

Miratur is:

• present tense
• indicative mood
• 3rd person singular
• from the deponent verb miror, mirari

So grammatically it means:

• he marvels at
• she marvels at
• it marvels at

In this sentence, because the subject is puer, it means the boy marvels at or the boy admires.

They describe where the fulgur is.

• procul is an adverb meaning far away, far off, at a distance
• supra montes means above the mountains

Together, they give the sense of something like:

• far off above the mountains
• far above the mountains

So procul does not take a case itself here; it simply adds the idea of distance to the whole spatial picture.

Because supra here is a preposition that takes the accusative.

So:

• montes = accusative plural of mons, montis = mountain

English speakers often expect a different case for location, but Latin prepositions each have their own case patterns. You cannot assume that every location word will use the ablative.

Here the correct construction is:

• supra + accusative
• supra montes = above the mountains

Because Latin has no articles.

That means Latin does not have separate words for:

• the
• a
• an

So puer can mean:

• the boy
• a boy

and fulgur can mean:

• the lightning
• a flash of lightning

The context tells you which is more natural in translation.

Latin word order is much freer than English word order because Latin uses case endings to show each word's role.

English depends heavily on position:

• The boy admires the lightning
• The lightning admires the boy

Those mean different things because the order changes.

Latin can move words around more easily because the endings already show who is doing what. In this sentence:

• puer marks the subject
• fulgur works as the object
• montes is governed by supra
• miratur comes naturally at the end, which is very common in Latin

So the order is not random, but it is less rigid than English.

Yes, in many cases they could, though the emphasis would change.

For example, Latin might also say:

• Puer procul supra montes fulgur miratur
• Fulgur puer procul supra montes miratur

The core meaning would still be understood from the endings, but the placement would shift attention or emphasis.

A few useful points:

• putting a word earlier can make it stand out more
• putting the verb at the end is very common
• keeping supra close to montes is usually clearer and more natural

So Latin word order is flexible, but not meaningless.

Usually fulgur suggests a flash or stroke of lightning rather than lightning as a completely vague mass.

So in this sentence, it is easy to imagine:

• a visible flash of lightning
• a bolt seen far above the mountains

That fits well with miratur, since the boy is gazing at a striking sight in the sky.

Latin words for weather and natural phenomena often overlap a bit in meaning, but here fulgur is a very vivid choice.