jmc

An integer that is congruent to $1(\mathrm{mod}23)$ is of the form $23n+1$.

Therefore, we form the inequality $23n+1<-999$, and find the largest possible integer $n$. We get $$\begin{array}{rl}23n+1& <-999\\ 23n& <-1000\\ n& <-\frac{1000}{23}\approx -\mathrm{43.48.}\end{array}$$ The largest possible negative integer $n$ is $-44$. We plug it in for $n$ to get $23\cdot -44+1=\boxed{-1011}$.