The basic data is usable in qgis.
Arcgis is the native software that shape files are used in.
Now shape files are in the open specification realm. There are many other data formats though that both qgis and arcgis read and write.
The only variable with shapefile transfer would be the coordinate system.
You might just ask the arcgis user what projection they normally use.
If you are only interested in transferring raw data to the Esri user then you are correct in your comment. The question seems to be answered by a comment.
My own reading of your question eludes to you wanting to transfer cartographic or other project configured information. i.e. the thematic settings of a data layer colors, thicknesses, visualisations; and possibly the map labeling, or other overall map conveyance.
I have seen similar questions asked about converting qgis project settings to other software.
These 2 searches show some of the relationships between the file types of the two software QGis asked here and ArcGIS.
QGIS uses qml files to apply layer configurations to other layers.
ArcGIS uses lyr files to apply layer configurations to other layers.
If this is not the type of information you are wanting to transfer with the project. Update (edit) your question with some description to clarify.