How to find the inverse function of f(x)=e^x+x-1

Roughly speaking, you can't, you don't.

You're not going to be able to put this in the form x = g(y) , (where y = f(x)).

The best that you can do is to form a table of values, and that will involve some sort of numerical procedure(s).

Here is the result given by Wolframalpha.........

[ Sorry.....I'm not familiar with the "Product Log Function,"   W  ........maybe some other mathematician on here knows what this is......  ]