I have a function that is
$$F_X(x) \in (0, 1]~~\text{for}~~0 < x_0 < x$$ and $$F_X(x) = 0~~\text{for}~~0 < x \leq x_0.$$
For $x > x_0$, $F(x)$ follows the properties of the CDF (e.g., right-continuous and non-decresing). Does $F(x)$ exist as a proper CDF at or below $x_0$? I am also unsure how to check whether $F(x)$ is right-continuous at $x_0$ or not.
In order that $F$ be right-continuous at $x_0,$ you need $\lim\limits_{x\,\downarrow\, x_0} F(x) = F(x_0).$ I.e. $x$ approaches $x_0$ from the right. You've decided that $F(x_0)=0,$ so you need $\lim\limits_{x\,\downarrow\,x_0} F(x) = 0.$ Whether that is true depends on what $F(x)$ is for values of $x$ that are greater than $x_0.$ And you have not told us what those are.