By "fancier lounge" you clearly mean the First Class lounge, which is indeed located on the floor above the Business Class lounge in Dubai Conccourse A (note if your flight departs from another concourse, such as C or D there are other lounges there).
The only quibble is regarding EK flights to South Africa, though the moe I look into this it seems that this may not pertain to the DXB lounge, but is related to QF elites trying to access the EK lounge in SYD when flying on QF flights to the US and JNB, which are not part of the QF/EK agreement.
If someone can find any T&C that indicates flights from DXB-South Africa are excluded for QF elites then feel free to post it for all our info
Otherwise I can't find anything to verify the previous post regarding this.
I think in DXB in both legs you should be given entry to the DXB First lounge (and it should state as such on your boarding pass)
In CPT it seems EK has a reasonably new lounge to enjoy too:
DUBAI, U.A.E., 23 MAY 2016 – Emirates’ premium and loyal customers can now look forward to even greater levels of comfort, hospitality and convenience with the opening of the airline’s new two-level premium lounge at the Cape Town International Airport in South Africa. All Emirates First Class...
www.emirates.com