The maths is different in non-euclidean geometry.
For example, go look at the earth's atlas. When you said that wrapping around from the south pole gets you to the north pole, that's wrong. The wraparound will keep you in the north pole except it will displace you 180 degrees to either side. For example if I am going up along the 20 degree longitude ill wraparound to the 180+20 = 200 degree longitude, then ill keep going forward and reach the south pole then wrap around back to 20 degree longitude.
Another thing is that the distance between the two ends keeps on changing depending on the latitude. If you are at the equator, and if you go east along the equator and come back to the same point you will travel 2*Pi*earth's radius distance. As you go up this distance(to keep going east until you reach the same point) keeps decreasing and at the poles, its 0.
So when you said you want to go east from the poles, you will keep standing at the same point because the pole's 'radius' is 0. (Because non-euclidean east is not the same as euclidean east)
What you actually meant to say was "I go to the north pole, turn 90 degrees to one side, then start walking" in which case, you will just switch longitudes.
So if you went to the pole along 20 degree longitude and turned 90 degrees and started walking, you would end up moving down along (20+90+180) = 290 degrees longitude. Moving along that longitude you eventually find the south pole as expected.
If you don't understand anything I am saying, get a globe of earth and an atlas of earth(which is essential a 2D representation of the earth - what you want to achieve) and move your finger around the globe and plot the path your finger takes on the atlas. Try to do the things you talked about(moving east from the pole etc.) and see how it plots on the atlas.