EDIT: The second tab of the find tool provide access to geocoding service online, so the more effective way to search places is to use such services, if the gazetteer in your area of interest is sufficient. Since it is not the case in my country, this option was not considered in the initial answer, which can provide help for those in the less lucky area.
As Fezter commented, you cannot search a raster map by name of cities in it.
If you want to, try digitizing your georeferenced map of cities to a point feature class. To do this, you should create a new feature class in a geodatabase (.gdb) or just create a shapefile (.shp).
After creating the feature class or shapefile, and before you create points, you may like to create a name field for this feature class. Right click on the layer in TOC, then select open attribute table. In the table dock, click on the first button on the top left of it, then add field. You may name it and choose its type as text.
After setting up the name field, you can select start editing from the Editing toolbar or right click menu of the layer in TOC. Then you'll be able to create points by clicking on the points in your map (for details, see https://blogs.esri.com/esri/arcgis/2010/07/28/getting-started-with-creating-features-in-arcgis-10/ ). And edit the name field according to your raster map.
When everything is done, save edits and stop editing. Then you can search your cities by the Find tool mentioned in another comment, or by Select by attribute tool in Selection menu.