You could also created a field in your polygon feature class (FC) with the value you want. Then perform an intersect between the point FC and the polygon FC. The resulting point FC will have the attributes of both. If you want this value back in your original point FC, then join the intersected FC with point feature class using a unique identifier (e.g. ID) and use the field calculator to make the value in the original FC equal to the value, then unjoin the FCs.
The spatial join would be simpler, but you still need to do the final step if you want to unjoin the FCs.