Question: How to write $f(x) = x^3 + 2x$ in 'completing the cubic' form [like we do in 'completing the square' for quadratic function]

Answer: 
You could write it as:

$x^3+2x=x^3+3x^2+3x+1-3x^2-x-1=(x+1)^3-3x^2-x-1$

I doubt, that there is a specific method to complete a cubic, in the sense, that you get it in the form $x^3+2x=a(x+e)^3+f$

Edit: In fact, if you write the equation above as

$x^3+2x=ax^3+3aex^2+3e^2x+e^3+f$

and now compare the coefficients on both sides, we see $a=1$. Also $e=0$, since there is no $x^2$ on the RHS. 
This leads to $x^3+f$ on the LHS, which is already wrong, so no such form exists here.
Depends what you mean.

Completing the square for a quadratic $f(x)$ means finding an expression
$$f(x)=(ax+b)^2+c\ ,$$
where $a,b,c$ are constants.

The obvious generalisation to a cubic is to look for $a,b,c$ such that
$$f(x)=(ax+b)^3+c\ .$$
If $f(x)=x^3+2x$ then expanding the right hand side and equating coefficients gives the equations
$$a^3=1\ ,\quad 3a^2b=0\ ,\quad 3ab^2=2\ ,\quad b^3+c=0\ .$$
But these equations have no solution ($3a^2b=0$ means that $a=0$ or $b=0$, then $3ab^2=0$ which contradicts $3ab^2=2$).

Therefore completing the cube in this sense is impossible for your example. It would be possible for certain examples, e.g.,
$$x^3+6x^2+12x+13=(x+2)^3+5\ .$$