Māori Grammar

Core meaning and function
Kātahi ka… is a very common sequencing marker in narratives and instructions. It signals that one event happens next, following what has just been mentioned: “and then / then / after that”. In this use, the verbal particle introducing the verb is typically ka. (Te Aka Māori Dictionary)

Typical pattern (very common)
Kātahi + (subject) + ka + verb …
Te Aka notes that the subject is often placed between kātahi and ka. (Te Aka Māori Dictionary)

Also possible
You will also see Kātahi ka + verb … (with the subject appearing later in the clause), especially when the subject is long/heavy, or for flow in storytelling. (This matches the pattern you quoted.)

Not the same as the exclamation “Kātahi rā!”
Kātahi can also be an interjection expressing surprise/disapproval (e.g., Kātahi rā hoki! “Good grief!”). That’s a different use from the sequencing kātahi ka…. (Te Aka Māori Dictionary)

Examples

Kātahi au ka tū.
Then I stood up.

Kātahi ia ka oma.
Then he ran.

Kātahi rāua ka kata.
Then they laughed.

Kātahi mātou ka kai.
Then we ate.

Kātahi koe ka kōrero.
Then you spoke.

Kātahi te kurī ka tangi.
Then the dog cried/whined.

Kātahi ka moe te pēpi.
Then the baby slept.

Kātahi ka kati te tatau.
Then the door closed.

Kātahi a Mere ka waiata.
Then Mere sang.

Kātahi te kaiako ka pātai.
Then the teacher asked (a question).

Kātahi ka horoi au i ōku ringa.
Then I washed my hands.

Kātahi ka hoki rātou ki te kāinga.
Then they went back home. (Te Aka Māori Dictionary)