I'm aware that there are a lot BBP type formulas out there which extract the n-th digit of the observed constant.
I'm asking for the reverse action, namely, is it possible to find the first occurrence of a given digit (or string of digits) within a constant? If not, any related work, analysis and other hints are welcome.
In short, there is no good way to determine the first occurrence of a string of digits within $\pi$. In fact, we do not know whether every finite string of digits actually occurs in $\pi$. [We believe this is true, but we do not know how to show it].
For some particular constants, it's rather easy. For instance, perhaps the simplest normal number is the number $0.12345678910111213\ldots$, formed by concatenating all base $10$ digits together. It should be pretty easy to determine the first occurrence of a string of digits here, but this is a very contrived example.