You cannot copy the Staffing Profile 1:1 I think, mainly because the resource pool assignments in respect to roles, resources etc might have changed and requests to the RP manager would have to be set up again.
You can 'Copy positions from exising Staffing Profile' and populate them again as before.
Action Plan = Workplan?
If so, when creating the project you can copy the workplan from another project, For obvious reasons, this can be done from the project that you are wishing to choose the staffing profile from. You then have the options what exactly what to copy from the previous workplan.
Staffing Profile is no problem, just re-link it to the new project no data is lost.
Workplan - 'copy workplan from another project' does the trick. If you want to include resources on each task you have to check what resource pools these users are registered in and temporarily set your self as an resource pool manager in these pools. Actuals is lost though, make sense.