$2023 \equiv 23 \pmod{m} \implies m | 2023 - 23 = 2000$ then subtract number of factors <=23

$$2023 = mn + 23, m > 23$$$$\implies mn = 2000, m>23$$$$2020 = 2^4 \cdot 5^3$$$2000$ has total $(4+1)(3+1) = 20$ positive factors, $8$ of which $(1, 2, 4, 5, 8, 10, 16, 20)$ are less than $23$. Thus, there are $12$ such numbers $(m)$ with the property.