You should use the Project Raster function with WGS1984 UTM Zone 37N as input coordinate system and Adindan UTM Zone 37N as output coordinate system.
The input coordinate system should be entered automatically if it is defined on the raster. However, the fact that they don't overlap in Arcmap suggests that's not the case for either your raster, shapefile or both.
You should also be aware of the difference between the Project and Define projection functions. The first one will change the coordinates of your file to a new a coordinate system while staying in the same spot on Earth. The latter will only write to your file which coordinate system it is already in (eg. where 0, 0 is), and wont change any of your coordinates (as they are stored in the file).