Asymptotic expansion for the inverse function of $ f(x)=x^3+x$
$g(x)$ is the inverse function for $f(x)$. the reciprocal would be $1/f(x)$.
To compute the inverse of $x^3+x$, let $$ y=x^3+x=x^3\left(1+\frac{1}{x^2}\right)\tag{1} $$ Taking $(1)$ to the $-\frac13$ power (via the binomial expansion) yields $$ \begin{align} y^{-1/3} &=\frac{1}{x}\left(1-\frac{1}{3x^2}+\frac{2}{9x^4}-\frac{14}{81x^6}+O\left(\frac{1}{x^8}\right)\right)\\ &=\frac{1}{x}-\frac{1}{3x^3}+\frac{2}{9x^5}-\frac{14}{81x^7}+O\left(\frac{1}{x^9}\right)\tag{2} \end{align} $$ Inverting the series for $u-\frac13u^3+\frac29u^5-\frac{14}{81}u^7$ yields $$ \begin{align} \frac1x &=y^{-1/3}+\frac13y^{-1}+\frac19y^{-5/3}+\frac{2}{81}y^{-7/3}+O\left(y^{-3}\right)\\ &=y^{-1/3}\left(1+\frac13y^{-2/3}+\frac19y^{-4/3}+\frac{2}{81}y^{-2}+O\left(y^{-8/3}\right)\right)\tag{3} \end{align} $$ Taking the reciprocal of $(3)$ yields $$ \begin{align} x &=y^{1/3}\left(1-\frac13y^{-2/3}+\frac{1}{81}y^{-2}+O\left(y^{-8/3}\right)\right)\\ &=y^{1/3}-\frac13y^{-1/3}+\frac{1}{81}y^{-5/3}+O\left(y^{-7/3}\right)\tag{4} \end{align} $$ The third term would normally be a $y^{-1}$ term, but the coefficient for that term is $0$.
In the derivation in the question, you get $$ g(x)=x^{1/3}-\tfrac13x^{-1/3}+h(x)\tag{5} $$ where $h(x)=o\left(x^{-1}\right)$. Using $(5)$, we get $$ \small g(x)^3+g(x)=x-\frac{1}{27}x^{-1}+\left(3x^{2/3}+\frac13x^{-2/3}-1\right)h+\left(3x^{1/3}-x^{-1/3}\right)h^2+h^3\tag{6} $$ Thus, we need to have $3x^{2/3}h=\frac{1}{27}x^{-1}+o\left(x^{-1}\right)$; that is, $h=\frac{1}{81}x^{-5/3}+o\left(x^{-5/3}\right)$.
Since the only powers of $x$ that appear in $$ \left(x^{1/3}-\tfrac13x^{-1/3}+\tfrac{1}{81}x^{-5/3}\right)^3+\left(x^{1/3}-\tfrac13x^{-1/3}+\tfrac{1}{81}x^{-5/3}\right)-x\tag{7} $$ are odd powers of $x^{1/3}$, the remainder, which is $o\left(x^{-5/3}\right)$, must be $O\left(x^{-7/3}\right)$
Thus, $$ g(x)=x^{1/3}-\tfrac13x^{-1/3}+\tfrac{1}{81}x^{-5/3}+O\left(x^{-7/3}\right)\tag{8} $$ which agrees with $(4)$.
The substitution $x = y^{1/3} u^{1/3}$ takes the equation $x^3 + x = y$ into $u + y^{-2/3} u^{1/3} = 1$. As in my recent answer to Solving a sum of exponentials, there is a series solution
$$ u = \sum_{k=0}^\infty \frac{(-1)^k a_k}{k!} y^{-2k/3} \ \text{where} \ a_k = \prod_{j=0}^{k-2} (k/3-j)$$ taking $a_0 = a_1 = 1$.
Now we have
$$x = y - x^3 = y - u y = \sum_{k=1}^\infty \frac{(-1)^{k+1} a_k}{k!} y^{1-2k/3} $$
$$ = y^{1/3}-\frac13 y^{-1/3} + \frac{1}{81} y^{-5/3} + \frac{1}{243} y^{-7/3} - \frac{4}{6561} y^{-11/3} - \frac{5}{19683} y^{-13/3} + \ldots$$
EDIT: The series converges for $y > 2 \sqrt{3}/9$.
Let $\color{blue}{z=x^{1/3}}$. Then $z^{-1}g(x)=1-\frac13z^{-2}+h(z)$ with $h(z)=o(z^{-2})$. Expanding the cube $(1+u)^3=1+3u+3u^2+u^3$ with $u=-\frac13z^{-2}+h(z)$ and neglecting every $o(h(z))$ and every $o(z^{-6})$ term in $u^2$ and $u^3$ yields $$ u^2=\tfrac19z^{-4}+O(z^{-2}h(z)),\quad\text{which implies}\quad u^2=\tfrac19z^{-4}+o(h(z)), $$ and $$ u^3=-\tfrac1{27}z^{-6}+O(z^{-4}h(z)),\quad\text{which implies}\quad u^3=-\tfrac1{27}z^{-6}+o(z^{-6}), $$ hence $$ z^{-3}g^3(z)=1-z^{-2}+3h(z)+\tfrac13z^{-4}+o(h(z))-\tfrac1{27}z^{-6}+o(z^{-6}). $$ Since $z^{-3}g^3(z)+z^{-3}g(z)=1$ and $z^{-3}g(z)=z^{-2}-\frac13z^{-4}+o(h(z))$, after some simplifications one gets the condition $$ 3h(z)-\tfrac1{27}z^{-6}+o(z^{-6})+o(h(z))=0. $$ Solving for $h(z)$ yields $h(z)=\tfrac1{81}z^{-6}+o(z^{-6})$, hence $$ \color{red}{g(x)=x^{1/3}-\tfrac13x^{-1/3}+\tfrac1{81}x^{-5/3}+o(x^{-5/3})}. $$
Edit (to answer a comment)
(i) The beginning of the expansion of $z^{-1}g(x)$ is $1-\frac13z^{-2}$, as you proved yourself, hence the remainder $h(z)$ may be any term $o(z^{-2})$. The idea of the exercise (and the reason why I had a look at it and finally wrote down a solution) is that one does not know a priori the order of $h(z)$. (As an aside, you might want to check Landau little-o notation.)
(ii) The term $3h(z)$ has nothing to do with the order of $h(z)$, simply the beginning of the expansion of $(1+u)^3$ is $1+3u$, here $u=-\frac13z^{-2}+h(z)$ hence $3u$ yields $-z^{-2}+3h(z)$.
(iii) Neither $\tfrac13z^{-4}$ nor $-\tfrac1{27}z^{-6}$ is $o(h(z))$, in the end. But it is not necessary to know this for the equality where they appear to hold.
(iv) The last equation is $3h(z)+o(h(z))=\tfrac1{27}z^{-6}+o(z^{-6})$, which is logically equivalent to the fact that $3h(z)=\tfrac1{27}z^{-6}+o(z^{-6})$.