I know you can specify code to be run in a wide variety of situations like when an exception is throw, method is executed, method is called, etc.
Is there any way that I can specify code to be run after every single line of execution of a given method?
No. How would AspectJ know how to do that? One line of Java code can be compiled into one or multiple Java bytecode statements. And why would you want to do this anyway? Better learn how to use a debugger.