Yes, it does sound natural. It is a quotation. You can see that from the use of the '"'. That means that it is a transcription of what someone spoke. You can also tell that from the 'oh...' and the way they almost repeat themselves, e.g. "When it's New Years" "when it's January 1st". When you speak, it is quite natural to make mistakes, or at least for what you say to sound like a mistake, when you write it down.

It is a bit awkward. I suggest: "in Canadian culture and many Western cultures, for New Year's Day, we often make resolutions. On January 1st, we make promises to ourselves that we're never going to do something again or that we're going to start doing something."