"the sun rises in the east, sets in the west and moves through the southern sky" That last part is dependent on your latitude, correct? If you were far enough south, the sun would move through the northern sky, wouldn't it?
When I did the same walk in February at the same time the sun was high on my left shoulder.
However assuming the sun does rise (i.e. you're not in a polar region near the solstices), it will always rise somewhere to the east (even if it's south-south-east) and set somewhere to the west.