Ég þarf að fylla út eyðublaðið áður en ég fer heim.

In Icelandic, þurfa (to need to) commonly governs an infinitive introduced by að. So the pattern is [subject] + þarf/þarfð + að + infinitive:

• Ég þarf að fylla… = I need to fill… You’ll see the same with many verbs that take an infinitive clause, e.g. Ég ætla að fara (I intend to go), Ég reyni að læra (I try to learn).

It’s a verb + particle construction, similar to English fill out. The core verb is fylla (fill), and út functions like a particle meaning “out/completely” in this idiom.

• fylla út eyðublað = to fill out a form
  In some contexts, the particle can move (especially with pronouns), but learning it as the fixed phrase fylla út is a good starting point.

Icelandic attaches the definite article to the end of the noun as a suffix.

• eyðublað = a form (an empty form/template)
• eyðublaðið = the form
  Here -ið is the definite ending for a neuter noun in the singular (nominative/accusative).

It’s the direct object of fylla út, so it’s in the accusative. For many neuter nouns, the nominative and accusative look the same, especially when definite:

• eyðublaðið can be either nominative or accusative by form alone
  You identify the case mainly by function: direct object → accusative.

Áður en is the standard conjunction meaning before when introducing a clause:

• áður en ég fer heim = before I go home
  Yes, en can also mean “than/but,” but in áður en it’s part of a fixed conjunction.

Icelandic normally requires an explicit subject in each finite clause, just like standard English:

• …before I go home (English keeps I too)
  You generally don’t omit the subject in Icelandic unless you’re using special constructions or very informal ellipsis.

The clause áður en ég fer heim is a subordinate clause, and Icelandic subordinate clauses typically have a more “straight” order: subject + verb (+ adverbs/objects). So ég fer is exactly what you expect. (The big word-order change learners notice is that in subordinate clauses, some adverbs tend to come before the verb more consistently than in main clauses.)

Heim is an adverb meaning home (towards home), and it often appears without a preposition:

• fara heim = go home
  If you mean “at home,” you usually use heima:
• vera heima = be at home

Both are present tense, 1st person singular:

• að þurfa → ég þarf (I need)
• að fara → ég fer (I go)
  Note that fara is irregular in the present tense (it doesn’t become fara → far the way some learners guess).

It’s very common and natural. You may also see:

• útfylla (eyðublað) (a single verb meaning “fill out”)
  But fylla út is everyday Icelandic and very widely used.

A rough guide (not perfect IPA, but learner-friendly):

• Ég ≈ “yehG” (with a palatal “y”-like onset; the g is soft/brief)
• þarf ≈ “tharv” (voiceless þ like English thing)
• eyðublaðið: the ð is like the th in this, and stress is on the first syllable: EY-ðu-bla-ðið
  Icelandic stress is almost always on the first syllable of a word.

You generally shouldn’t drop að after þarf in standard Icelandic; it’s the normal construction. Reordering is possible for emphasis, but the given sentence is already the most neutral and natural:

• Ég þarf að fylla út eyðublaðið áður en ég fer heim. (neutral, standard)