I'm using an approximation of $x/\ln(x)$ ~ $π(x)$ and I am using the logic that the square root function counts how many squares there are and the log function counts how many powers of base b there are, and inverting those functions gives you the nth square or power. The Pi function works the same way as the other listed functions, where the value increases by 1 every new prime, and the same with the approximation, so does it make sense to take the inverse of $f(x)=x/\ln(x)$, which is $f^{-1}(x)=-xW_{-1}(-1/x)$ as the approximation for the nth prime? If so, how good of an approximation is this?
(I used only the branch of $W_{-1}$ which would experience a positive, unbounded growth since using $W_0$ would have given me only values below 2 which I didn't want.)
The approximation is asymptotically correct. That is $$p_n \sim -n\,W_{-1}(-1/n)$$
but that's not a very good approximation. That's because $\frac{x}{\log x}$ is not a very good approximation of $\pi(x)$ to begin with.
A better approximation of $\pi(x)$ is $\frac{x}{\log x -1}$.
If you invert that you get a much better approximation:
$$p_n \sim -n\,W_{-1}(-e/n)$$