Hun er på reise i dag, så jeg passer på katten hennes.

Å være på reise is a common Norwegian expression meaning to be traveling / to be away on a trip (focused on the state of being away).
Hun reiser i dag usually means she is traveling today in the sense of the act (often implying departure or travel happening today).
So:

• Hun er på reise i dag = she’s away traveling today (status)
• Hun reiser i dag = she travels/leaves today (event)

In Norwegian, the idiom is på reise (literally on a journey). Norwegian uses på in several “activity/state” expressions: på jobb, på ferie, på besøk, på tur, på reise.
I reise is generally not used in this meaning.

Adverbs of time like i dag often come after the verb phrase they modify:

• Hun er på reise i dag.

You can also move time to the front for emphasis, but then word order changes:

• I dag er hun på reise. (fronted time → verb comes before subject)

Here så functions like so / therefore to show a consequence:

• Hun er på reise i dag, så jeg passer på katten hennes.

It’s very common in spoken and informal written Norwegian. A more formal alternative is derfor:

• Hun er på reise i dag, derfor passer jeg på katten hennes.

Norwegian is generally V2 (verb-second) in main clauses, but after så used as a coordinating linker (similar to and/so), the next clause is typically treated like an independent main clause with normal order:

• …, så jeg passer … (subject jeg
  • verb passer)

If you use a subordinating conjunction like fordi (because), you’d get subordinate clause order:

• … fordi hun er på reise i dag. (often places negation/adverbs before the verb in the subordinate clause)

Å passe på is a fixed verb + particle expression meaning to look after / take care of / keep an eye on.
It’s not the same as å passe alone, which can mean things like to fit / to suit:

• Skoene passer. = The shoes fit.
• Jeg passer på katten. = I’m looking after the cat.

The på belongs to the expression, so you normally keep it.

Usually you should keep på.
Å passe noen/noe can exist, but in modern everyday Norwegian the most natural for “look after” is passe på:

• Most natural: Jeg passer på katten hennes.
• Possible but can sound less standard/less precise: Jeg passer katten hennes.

Katten is the definite form (the cat), and it’s natural because the speaker and listener typically know which cat (her cat). Norwegian often prefers definite forms when referring to a specific, known thing:

• katten hennes = her cat (a specific one)

En katt would sound like “a cat (some cat)” rather than the known pet.

Norwegian commonly places possessives after the noun: noun + possessive. This is very frequent in everyday language:

• katten hennes = her cat
• bilen min = my car

You can place the possessive first, but then the noun is usually indefinite:

• hennes katt (more formal/emphatic; often used to contrast whose cat it is)

So:

• katten hennes = the cat of hers (neutral, common)
• hennes katt = her cat (contrast/emphasis)

Use hennes when the owner is someone else (not the subject of that same clause). Use sin/sitt/sine when the owner is the subject of the clause.

In your sentence:

• … jeg passer på katten hennes.
  The subject is jeg, so the cat belongs to her, not to me → hennes.

Compare:

• Hun passer på katten sin. = She looks after her (own) cat.
• Hun passer på katten hennes. = She looks after her cat (someone else’s cat).

It’s standard main-clause order: 1) Subject: jeg
2) Verb: passer
3) Particle/preposition: på
4) Object: katten hennes

So: S + V + (particle) + O.

Because it separates two main clauses:

• Hun er på reise i dag,
• så jeg passer på katten hennes.

In Norwegian, it’s normal to use a comma before connectors like så when they link two independent clauses.

Two common tweaks:

• Replace så with derfor:
  • Hun er på reise i dag, derfor passer jeg på katten hennes.
• Or split into two sentences:
  • Hun er på reise i dag. Jeg passer på katten hennes.

Both feel more formal than the very conversational …, så … structure.