jmc

Notice that $1001=10\cdot 100+1$. Therefore, $$10\cdot 100\equiv -1(\mathrm{mod}1001),$$which implies that $$10\cdot -100\equiv 1(\mathrm{mod}1001).$$The inverse of $10(\mathrm{mod}1001)$ is $-100$, but we need an answer in the interval from $0$ to $1000$. The equivalent residue in this interval is $-100+1001=\boxed{901}$.

We can check our answer: $10\cdot 901=9010=9(1001)+1\equiv 1(\mathrm{mod}1001)$, so $901$ is indeed the inverse of $10(\mathrm{mod}1001)$.