ArcMap can reproject the sid on the fly if it to match your other data IF it has the correct world file information (either in a world file or in the side header) and IF it has the correct coordinate system definition (in the header or in another file, aux.xml is the one ArcMap creates). You need both.
If you just lack a coordinate system use the Projection Define tool or in ArcCatalog the right click - Properties - Coordinate System tab to give it the correct definition. You don't reproject a sid by modifying the world file or projection definition file directly. After you properly define a coordinate system, you could use the Project tool to change it, but since you can't save the new raster as a sid, and since ArcMap can reproject on the fly, you probably won't want to do that.
If your sid has not been georeferenced, you can do that with the georeferencing toolbar, and that will create a world file and a projection definition. See the ArcGIS help on geoerferencing.
