The eye dropper tool works wonders if you don't know what the color is but want to figure it out.
http://www.esri.com/esri-news/arcwatch/1214/finding-colors-on-maps-is-easy-using-the-eye-dropper-tool
The trick in your case is that there is an underlying hillshade that is interacting with the colors. I'd pick a low elevation and a high elevation with relatively moderate hillshade values. Use the eye dropper tool to get the RGB values. In ArcGIS you can set the two end colors in a color ramp by right clicking on the color ramp of the layer in the Table of Contents (left panel where all of the layers are listed). If your attached map isn't a georeferenced layer you can still bring it into ArcGIS as an image and use the eye dropper, but you'll need to manually guess which areas have high and low hillshade values.
Alternatively, just e-mail the authors if you can find their names and ask.