Create a text field, let's say it's called "PrintDate". Then go to its Properties and under the General tab select "Form Field: Hidden but Printable".
Now go to Tools - JavaScript - Document Actions and enter the following code as the Will Print event:
this.getField("PrintDate").value = util.printd("mm-dd-yyyy HH:MM", new Date());
This will cause the field to update with the current date and time just before the file is printed.