Jag måste skriva upp adressen innan jag går hem.

Because after the modal verb måste (must / have to), Swedish normally uses the infinitive without att.

• Jag måste skriva. = I must write.
• Hon kan komma. = She can come.
• Vi vill äta. = We want to eat.

So måste skriva is the normal pattern.

Skriva upp is a particle verb and here it means write down, note down, or make a note of.

Compare:

• skriva = write
• skriva upp = write down / note down

So:

• Jag måste skriva upp adressen means you need to make a note of the address, not just generally write.

This is similar to English phrasal verbs like write down, where the extra word changes the meaning.

Adressen is the definite form of adress, meaning the address.

• en adress = an address
• adressen = the address

The sentence uses adressen because it refers to a specific address, one that is already known from the situation.

Swedish often adds definiteness directly to the noun:

• en bok = a book
• boken = the book

So:

• skriva upp adressen = write down the address

Because innan jag går hem is a subordinate clause, and it needs its own subject.

The sentence has two clauses:

• Jag måste skriva upp adressen = main clause
• innan jag går hem = subordinate clause

Even though both clauses have the same subject in English and Swedish, Swedish still normally repeats it:

• Jag måste skriva upp adressen innan jag går hem.

You cannot usually leave out the second jag.

In Swedish, hem by itself often means homeward / home after verbs of movement.

So:

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

You usually do not say gå till hem.

A very useful contrast is:

• hem = motion toward home
• hemma = at home

Examples:

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

Because Swedish often uses the present tense for actions in the future when the time is clear from context.

So innan jag går hem literally uses present tense, but it means:

• before I go home

This is very similar to English, which also says:

• before I go home not
• before I will go home

So the Swedish tense here is completely normal.

Yes. Innan introduces a subordinate clause, and subordinate clauses in Swedish do not use the usual main-clause V2 word order.

That is why you get:

• innan jag går hem

with the subject jag before the verb går.

This is the normal subordinate-clause pattern.

A useful comparison:

• Main clause: Jag går hem nu.
• Subordinate clause: ...innan jag går hem.

If you add a sentence adverb, you can see the subordinate word order even more clearly:

• innan jag inte går hem would place inte before the verb, because it is a subordinate clause structure.

So innan jag går hem is exactly what you should expect after innan.

Often yes, and in many situations both can mean write down the address.

But there can be a slight nuance:

• skriva upp often suggests making a note of something, sometimes on a list or so you do not forget it
• skriva ner also means write down, often focusing on putting spoken or remembered information into writing

In everyday conversation, both may work, but skriva upp adressen sounds very natural for noting down an address.

Måste is a modal verb. Modal verbs are verbs like can, must, may, want to, and they are followed by another verb in the infinitive.

Examples:

• Jag måste skriva. = I must write.
• Hon kan läsa. = She can read.
• Vi vill gå. = We want to go.

So in your sentence:

• måste expresses necessity
• skriva upp is the action that must be done

Yes:

• Jag = I
• måste = must / have to
• skriva upp = write down / note down
• adressen = the address
• innan = before
• jag = I
• går hem = go home

That makes the structure easy to see:

I must + write down the address + before + I go home.