The graticule on the map suggests a conformal conic projection was used, because
meridians and lines of latitude always meet at right angles, indicating conformality
lines of latitude appear approximately circular and approximately equally spaced, characteristic of a conic projection.
I therefore attempted to register a screenshot of the map to a base feature projected using the "GDA 1994 Geoscience Australia Lambert" coordinate system (WKID 3112, EPSG). This is a Lambert Conformal Conic projection with central meridian at 134.0 degrees, standard parallels at -18 and -36 degrees, and meters for the units of measurement. The registration (an affine transformation) has a root mean square accuracy of 3 kilometers. How good is this? Some considerations are:
The map appears to be a scan of a paper copy. This will introduce some relatively large distortions that cannot be corrected.
It is an older map (1974) and therefore might not have been very accurate originally.
My screen shot has a resolution of only 35 pixels per degree (because it had to include the entire map). The graticule is still several pixels thick at this resolution and therefore we should be lucky if it is accurate to better than about 1/35 degree vertically = 3 kilometers.
Registration with a second-degree polynomial does not significantly decrease the root mean square residual (it decreases from 2720 meters to 2010 meters but requires an additional three parameters for the quadratic terms). This indicates the projection is either correct or close to correct (up to a simple change of origin as captured by a "false easting" and "false northing" value, or possibly a fixed rotation as reflected by the central meridian): using the wrong projection would cause the second-degree fit to be a significant improvement over the original affine fit.
I conclude provisionally that the original image likely can be registered to the EPSG 3112 coordinate system to within an acceptable error, given the qualities of the original map (and given that the essential features on it are only roughly drawn anyway: their boundaries are four pixels thick, or about 10 - 15 kilometers).
(I had difficulties with registration along the middle of the top of the map and attribute that to a local error in the graticule. Likewise, some of the graticule in the interior of the country may be grossly misplaced.)
This screenshot of the registration shows a five-degree graticule (black) and a polygon shape for Australia (gray with yellow outline) overlaid on the provisionally registered image of the original map. The red numbered crosses surrounding the figure are registration points: each one corresponds to a whole number of degrees in latitude and longitude and therefore should lie exactly on a crossing of the original map's graticule (which has a two degree spacing).