Here's one way to find a similar CRS to the one used for an existing map. Since your map has lat/long lines, you can make a GIS project with the same lines into different CRS's until you find one where the lines are curved and angled the same as in the image.
- Create lat/long lines:
- Use the grid tool to make a lat/long grid for this area (longitudes 23 E to 48 E, latitudes 32 N to 45 N) in EPSG:4326 (WGS84 unprojected).
Densify the grid, so it has lots of vertices. This way when you project it into another CRS, the latitude lines will appear curved.
Optional: add a vector layer of country boundaries for the area of interest.
- Eg, get a worldwide layer from Natural Earth Data, select the countries in the area of interest, and export them to a new layer.
Change the project CRS until you find a CRS where the grid lines are curved similarly to the ones in your map image. As @Cyril suggested, start with the conic projections. When you select a CRS in the CRS selector window, notice that the Extent is displayed below. Try to find one with an similar extent to your map.
Keep in mind that you don't have to find exactly the right CRS. A reasonably similar CRS will give you reasonably good results when you georeference the image. And you can always re-run the georeferencer if you're not happy with the results.
Tip: when you georeference the image, save the georeference control points (GCP) in case you want to re-run the georeferencer.