It's an idiomatic expression "only take a minute" often means "will not take very long" rather than the literal meaning. A literal translation here is more like "If I'm only running one kilometer, it will be immediate", which is also idiomatic. There is no literal 1 km in 1:00 being implied though.
My body... is a bit weird...I-it's ok!I-I'm telling you, you've got to put a towel on!Running 1K will only take a minute!*tremble**tremble**tremble**tremble*