You would ask to view the apartment with something like:
Ich würde gerne einen Besichtigungstermin vereinbaren.
Ist es möglich, dass ich mir die Wohnung einmal anschaue?
There is a wide variety of possibilities, depending on the grammatical context of the sentence and what else you want to communicate on a pragmatic level (e.g. correctness, friendliness, social skills, high income, etc.). As a non-naive speaker I would not worry about this aspect of language and explain that German is not your mother tongue.
I wouldn't attempt to communicate in a dialect such as Swiss German. Many dialects are quite different from Standard German and almost unintelligible for native German speakers unfamiliar with them. Standard High German was invented as a lingua franca for speakers of different German dialects, and all Swiss speakers understand and speak Standard German.
That said, a Swiss person might say something like: "Chan i d'Wohnig ahluëge?" If you don't know how to pronounce that, you shouldn't use it. Also, dialects aren't usually written (in fact I made up the spelling, as there is no normative orthography) and never used in formal writing, so as an applicant for an apartment you wouldn't use it even if you were Swiss.