Your basic problem is that the map you are attmepting to use is an image file that is not georeferenced. So unless you want to go through the unnecessary and probably time consuming process of georeferencing this image yourself, you will be better taking an alternative approach. There are perhaps a few ways to do this. But, unless you have very few data points to overlay on the map which you can place manually using the lat-long grid of the image, then the least painful method will certainly be to redraw the map yourself using the shapefile.
This is not the right place to give you an introductory lesson on GIS, but the basic steps are to
- Download shapefile (which is available at the same website as the image you linked)
- Project map to desired coordinate system
- Plot map, coloring by climate class
- Color the ocean layer
- Add labels, legend, and graticule, as desired
- Overplot with your own climate data, and legend for these.
If you are unsure how to approach any of these steps, then take an introductory course on GIS, and search the Web for instructional materials. You may find this resource useful.
https://cran.r-project.org/doc/contrib/intro-spatial-rl.pdf