Nouvelle-Calédonie - Part 1— partie 1/première partie

"une part" is a part from something to be shared, like a slice of cake, for instance
In every other case, it's "une partie"

J'étais là pour une semaine.

This sentence isn't wrong, but would rather mean "I was meant to stay/I intended to stay for a week"
The neutral way to say you spent a week there would be "j'y ai passé une semaine"

C'était génial, une très beau payslle île.

It's part of France, so technically not "un pays"
You may however say "une belle région". "Région" has a strict administrative meaning, but also a broader one

J'ai été surpris par les grandhautes montagnes et les nombreuses couleurs du paysage.

J'écirai quelques entrées de journal pour raconoter mon tempsséjour en Nouvelle-Calédonie.

"temps" isn't used as much as "time" is in English. Here, it would sound like an anglicism
un séjour = a stay, a journey