Let's analyze the problem step by step.

1. Square Properties: Since $ABCD$ is a square, all sides are equal, and the diagonals bisect each other at right angles. Let the side length of the square be $s$. Diagonal $BD$ would then have a length $BD = s\sqrt{2}$.

2. AX and CX Relationship: $AX$ meets diagonal $BD$ at point $X$. Since $AX = 2022$ cm and the question is asking for the length of $CX$, we can use the fact that $AX + CX = AC$, where $AC$ is the diagonal of the square.

3. Diagonal Calculation: The length of diagonal $AC$ is given by $AC = s\sqrt{2}$, and in a square, diagonals are equal, so $AC = BD$.

4. Proportionality: In a square, if a point $X$ lies on the diagonal $BD$, then the triangles $\triangle AXD$ and $\triangle CXB$ are similar by AA similarity (since they share angles).

   This similarity implies that: $\frac{AX}{CX} = \frac{AD}{BC} = 1$

   Therefore, $AX = CX$.

5. Conclusion: Given $AX = 2022$ cm, by symmetry $CX$ must also be $2022$ cm.

So, the correct answer is 2022 cm.

Would you like further details or have any other questions? Here are some related questions to consider:

1. What is the formula for the diagonal of a square?
2. How does the Pythagorean theorem apply to squares?
3. What are the properties of similar triangles?
4. Can we apply coordinate geometry to solve this problem differently?
5. How do the properties of a square help in solving this problem?

Tip: Always remember that in a square, the diagonals bisect each other and are equal in length. This symmetry is often crucial in solving problems involving squares.