expression
before ... days (months, etc.) have passed
Fixed expression meaning 'before a certain amount of time has elapsed'. Used with a time expression (e.g., 三日, 一か月) to indicate that something happens very soon, often unexpectedly quickly.
彼は入社して三日も経たぬうちに辞めてしまった。
He quit before even three days had passed since joining the company.
一か月も経たぬうちに、また同じ問題が起きた。
Before even a month had passed, the same problem occurred again.
経つ is the verb 'to pass (time)'; 経たぬうちに is a fixed negative form meaning 'before it has passed'.
うちに means 'while' or 'before (a state changes)'; combined with 経たぬ it creates 'before (time) elapses'.
From the negative form of 経つ (to pass, elapse) + うちに (while, before). Literally 'while (time) has not passed', i.e., 'before (time) has elapsed'.