Learn Icelandic: Eftir ferðalagið er barnið um það bil að sofna. - After the journey, the child is just about to fall asleep.

Literally, um það bil means "around that interval" or "approximately." When combined with the verb vera (here as er) and the infinitive marker að, it acts just like the English phrase "to be about to" do something. It shows that the action of falling asleep is imminent.

Why do we use the verb sofna instead of sofa here?

Sofa describes the ongoing state of sleeping, whereas sofna describes the transition of falling asleep. Since the child is "about to" enter that state, you must use the transitional verb sofna.

Why does the verb er come before the subject barnið?

This is a great example of the V2 (verb-second) rule in action. Because the sentence starts with the prepositional phrase Eftir ferðalagið, the main verb er must take the second structural position, pushing the subject barnið right after it.

What case is ferðalagið in, and why?

It is in the accusative case. When the preposition eftir is used to mean "after" in a temporal sense (referring to time), it governs the accusative. Because ferðalag is a neuter noun, its accusative form looks exactly the same as its nominative form.