To complete the task in ArcMap:
Just a note - Do Not Use ESRI's Default Lo27 Spatial Reference
Load all your points into Excel / A text editor. Make sure you have at least 3 columns. 1 - ID or Name for the point ; 2 - Latitude ; 3 - Longitude.
Save the file and open ArcMap.
Add the data to ArcMap as you normally would, right click the file and select "Display XY Data". The next window that will pop up allows you to select the X Field (Your Longitude Column), the Y Field (The Latitude Column) and a Z Field - leave blank, unless you have a Z value. The next important thing is to select the "Edit" button. This will let you select the coordinate system for your GPS Points. You stated that you have the data in "WGS84".
On pressing edit, you will be taken to the "Spatial Reference Properties" page. From the window with all the folders, select "Geographic", "World" and then "WGS84". Press ok, and then Ok again from the "Display XY Data" Window. Below is just a screenshot of the Spatial Reference Window.
You data view should now show the points as a layer. THis layer is temporary, so if you close the MXD you will lose it. It will have the WGS84 Spatial Reference. You now need to re-project this data to the Cape Lo27 reference. To do this, open the ArcToolbox, Navigate to "Data Management Tools" then "Projections and Transformations" and the "Project" Toolbox. See below:
Select th einput as you XY Event Layer, Choose a filename to save the Event Layer a permanent Shapefile. Now select the little button to the right of the "Output Coordinate System" to make a new coordinate system for CAPE Lo27. A new window will pop up, it will be the "Spatial Reference Properties" again. This time select the drop down arrow from the little earth button - Shown in the pic below. Select "New" -> "Projected Coordinate System"
Things can get a little tricky here.
First: Choose a name for the new system you want to make: Cape_Lo31
Second: Choose the projection system you want to use: Select Transverse MErcator from the drop down list
Third: Type 27 in space next to "Central Meridian"
Fourth: Linear unit is "Meters" and the Meters per unit is "1"
Fifth: Select "Change" in the Geographic Coordinate System
In this new window that pops up, scroll to the top, choose the "Geographic Coordinate Systems" folder, then "Africa" and search for "Cape". Select it.
Press ok, and finish running the "Project" Tool.
In the data view there will now be a point file, correctly projected in Cape Lo27.
To show the coordinates in the attribute table you need to change the Data Frame Property's Coordinate System to the new Lo27 system you have created. Right click the "Layers" at the top of the Table of Contents, Select Coordinate System from the tabs and you will see your new system under the "Layers" folder. Select it and then open the attribute table in your new point file.
Add two new fields "X_lo27" and "Y_lo27". Right click on the field and select "Calculate Geometry". For the X field, select the X Coordinate and for the Y field select Y.
When this process has run you can export this attribute file out into text for use in any other software you need to.