Both once and one time can be used to mean formerly and refer to an extended time in the past (seen as a single span of time). The OED offers more than one example, including the following:
He used to read one time. Doesn't he now?
(1971, Whistle in the Dark by T. Murphy)
There used to be a public-house there one time...
(1888, W. Somerset Word-bk by F. T. Elworthy)
In these uses, one time means formerly and not on a single occasion.
So, one time can indeed mean "a week or a month" (to quote your question)–just as once can.
Once/one time, I spent every day sending letters, pretending to be somebody I wasn't.
So yes you can use one time. However, such words as once and formerly are used more often in today's English (these examples with one time might seem a little odd to today's native reader).
As for the date of A Whistle in the Dark, the OED lists the usage as from 1971, a date confirmed by the World Catalog; this work does not have to be in exact accordance with the script of the play apparently performed ten years earlier, and in any case, 1961 is still contemporary English and indeed Drama Online considers Thomas Murphy a "contemporary dramatist."