If $673 < m < 1000$ then dividing $2019$ by $m$ gives quotient $2$ and remainder $2019 - 2m$. Obviously the remainder increases when $m$ decreases. Thus in the case $m = 674$ we obtain the largest remainder $671$.

Dividing $2019$ by $673$ gives remainder $0$. Dividing $2019$ by $672$ or any smaller number gives remainder that does not exceed $671$. Consequently, the largest remainder under the given conditions is $671$.