I created a QGIS shapefile layer in QGIS that I want to give to a friend of mine for his ArcMap project. Will he be able to open and use my work?
You need to understand that there is no such thing as a
QGIS Shapefile, A Shapefile is a shapefile regardless of what software was used to create it.
According to ArcGIS Online Help:
A shapefile is an Esri vector data storage format for storing the location, shape, and attributes of geographic features. It is stored as a set of related files and contains one feature class.
But as PolyGeo said, you will not be able to save the symbology directly. The layer files don't work across QGIS and ArcMap directly ( maybe there is a workaround, but I'm not sure)
If you give your friend the shapefile, and it contains a suitable *.prj, then it should be usable in ArcMap but will need to be symbolized anew. The best thing will be to perform a test transfer of data and see if it works for you.
If you are planning to share the shapefile and a layer definition file (*.qlr) to try and preserve its layer properties I would not expect the latter to be usable.