Can you invoke an instantiated object's class constructor explicity in C++?

You can use placement new, which permits

new (&instance) A(2);

However, from your example you'd be calling a constructor on an object twice which is very bad practice. Instead I'd recommend you just do

A instance(2);

Placement new is usually only used when you need to pre-allocate the memory (e.g. in a custom memory manager) and construct the object later.

No.

Create a method for the set and call it from the constructor. This method will then also be available for later.

class A{
    A(int a) { Set(a); }
    void Set(int a) { }
}

A instance;

instance.Set(2);

You'll also probably want a default value or default constructor.

No

Calling instance.A() or A(1) is seens as casting  'function-style cast' : illegal as right side of '.' operator

Usually if a function/functionality is to needed in constructor as well as after object is construted it is placed in init() methode and used in constructor and in other place too.

example:

 class A{
      A(int a)
       { 
        init(a);
       }

     void init(int a) { } 
     }

        A instance;

        instance.init(2);