$j(\tau)=J(e^{2i\pi \tau})$, since $J(q)\in q^{-1}+\Bbb{Z}[[q]]$ then $J^{-1}(1/s)=\sum_{k\ge 1} d_k s^k$ is in $\Bbb{Z}[[s]]$.
$$f(q)=1/J(q) = \exp(-2i\pi j^{-1}(\tau))$$ is surjective from $|q|< 1$ to $\Bbb{C}$ and locally biholomorphic away from $f(q)=1/1728$ and $f(q)=0$. In the branch where $f^{-1}(0)=0$ then $f^{-1}$ is analytic for $|q|<1/1728$.
$f'(1/1728)=\sum_{k\ge 1} \frac{k\, d_k}{1728^{k-1}}$ doesn't converge, this follows from the that $J'(i)=0\implies \lim_{s\to 1/1728^-}f'(s)=\infty$.
To get an upper bound on the $d_k$, let $m(y)=\sup_{\Im(\tau)>y} |1/j(\tau)|$. For $|s|<m(y)$ we'll have $J^{-1}(1/s)<e^{-2\pi y}$ so that $|d_k| \le m(y)^{-k-1} e^{-2\pi y}$ whenever $m(y)< 1/1728$. A numerical check shows that $m(2)< 10^{-5}$.
The monodromy group of $j^{-1}(\tau)$ is interesting:
in the branch where $j^{-1}(\infty)= i\infty$ when rotating around $\infty$ it transforms $j^{-1}(\tau)$ to $j^{-1}(\tau)+1=\pmatrix{1&1\\0&1}j^{-1}(\tau)$.
In the branch where $j^{-1}(1728)=i$ when rotating around $1728$ it transforms $j^{-1}(\tau)$ to $\pmatrix{0&1\\-1&0} j^{-1}(\tau)$.
In the branch where $j^{-1}(0))=e^{2i\pi/3}$ when rotating around $0$ it transforms $j^{-1}(\tau)$ to $\pmatrix{&1\\-1&1} j^{-1}(\tau)$.
In the branch where $j^{-1}(\infty)=\pmatrix{a&b\\c&d}i\infty=a/d$ it gets different, rotating around $\infty$ transforms $j^{-1}(\tau)$ to $\pmatrix{a&b\\c&d} \pmatrix{1&1\\0&1} \pmatrix{a&b\\c&d}^{-1} j^{-1}(\tau)$. Whence in the branch such that $f(q)=\exp(-2i\pi a/d)$ then $f(q) = \exp(-2i\pi j^{-1}(\tau))$ isn't analytic at $0$ anymore.