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I'm not sure this answers your question.
The time an object crosses the meridian (and achieves maximum altitude) is when the local sidereal time is equal to its RA. Just about any planetarium program will tell you the UTC or civil time when this happens. At that time (assuming the southern hemisphere) its azimuth is 180 (ie due south). That is when it changes from moving up to down.
Around the SCP, objects move clockwise, so when crossing the meridian it will be moving right.
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