Jag var tvungen att gå hem tidigt eftersom jag hade ont i huvudet.

Because this sentence is talking about a past obligation: I had to...

Swedish often uses vara tvungen att + infinitive for that:

• Jag är tvungen att gå = I have to go
• Jag var tvungen att gå = I had to go

Måste can sometimes be understood as past from context, but var tvungen att is usually clearer and very natural when you want to say had to.

Yes, tvungen changes form to match the subject.

Here it is tvungen because jag refers to one person.

Common forms:

• tvungen = singular common gender/person
• tvunget = singular neuter
• tvungna = plural

Examples:

• Jag var tvungen att gå.
• Barnet var tvunget att gå.
• Vi var tvungna att gå.

So in this sentence, jag var tvungen is the expected form.

Because the pattern is:

vara tvungen att + infinitive

So:

• Jag var tvungen att gå
• Hon är tvungen att jobba
• Vi var tvungna att vänta

The att is part of the construction. Some Swedish verbs do not use att before another verb, but this expression does.

Because hem often works like an adverb meaning home or homeward.

So Swedish says:

• gå hem = go home
• åka hem = go home
• komma hem = come home

You do not normally use a preposition here.

Also, notice the difference:

• hem = to home / homeward
• hemma = at home

So:

• Jag går hem nu. = I’m going home now.
• Jag är hemma nu. = I’m at home now.

Because tidigt is an adverb here, and it describes the verb gå.

• tidig = early, as an adjective
• tidigt = early, as an adverb

Compare:

• ett tidigt möte = an early meeting
• Jag kom tidigt. = I arrived early

In your sentence, tidigt tells us how/when the person went home, so the adverb form is needed.

Eftersom is a conjunction meaning because or since.

It introduces the reason:

• Jag var tvungen att gå hem tidigt eftersom jag hade ont i huvudet.

A very close alternative is därför att:

• ...därför att jag hade ont i huvudet.

Both are normal. Learners are often told to be careful with för att, because that very often means in order to, not just because.

Ha ont is a very common Swedish expression meaning to be in pain or to hurt.

Examples:

• Jag har ont. = I’m in pain.
• Jag har ont i ryggen. = My back hurts.
• Jag hade ont i huvudet. = I had pain in my head / I had a headache.

So ont is part of a fixed expression here:

• har ont = have pain / hurt
• hade ont = had pain / hurt (past)

With pain inside a body part or in that area, Swedish usually uses ont i + body part.

So:

• ont i huvudet = pain in the head
• ont i magen = stomachache / pain in the stomach
• ont i ryggen = back pain

På huvudet would sound more like something is on the surface of the head or on top of the head, not the usual way to describe a headache.

This is very common in Swedish. With body parts, Swedish often uses the definite form where English uses a possessive.

So Swedish prefers:

• Jag har ont i huvudet.
• Hon tvättade händerna.
• Han bröt benet.

English would usually say:

• my head
• her hands
• his leg

In Swedish, the possessor is often understood from the subject, so huvudet sounds natural. You could say mitt huvud, but that would usually sound more emphatic or contrastive.

They are very close in meaning.

• Jag hade ont i huvudet = I had pain in my head
• Jag hade huvudvärk = I had a headache

Both are natural. The first is a bit more literal and everyday; the second names the condition directly.

Also notice that Swedish usually says:

• ha huvudvärk

without an article. So jag hade huvudvärk is more natural than jag hade en huvudvärk in a simple statement like this.

Because eftersom introduces a subordinate clause, and Swedish subordinate clauses normally keep subject + verb order.

So:

• eftersom jag hade ont i huvudet

not:

• eftersom hade jag ont i huvudet

This is different from main-clause word order in Swedish, where the verb often comes earlier because of the V2 rule.

For example:

• Jag var tvungen att gå hem tidigt eftersom jag hade ont i huvudet.

If you start with the eftersom clause, the main clause changes word order, but the subordinate clause does not:

• Eftersom jag hade ont i huvudet, var jag tvungen att gå hem tidigt.