I consider a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ and I know that function $f$ is continuous at some $x_0=(y_0,z_0)$. Are there any other properties of function $f$ that can help me to conclude that function $f$ is in fact continuous in a neighborhood of $x_0$? I know, e.g., that function $f(y,z)$ is continuous from the right in $y$ for any $z$ and is continuous from the right in $z$ for any $y$. I also know that $f$ is non-decreasing in each component.
I am not interested in obvious properties that immediately imply the continuity of $f$ everywhere (such as differentiability, etc.).
I would argue that there is no such condition that doesn't feel unnecessarily strong.
Here are some examples:
The function defined by $$ f(y,z) = \begin{cases} 0,&z\in\mathbb Q\\ y,&z\notin\mathbb Q \end{cases} $$ is continuous at the origin and continuous in $y$ for any fixed $z$, but it is not continuous in any neighborhood of the origin. This does not satisfy your assumptions yet, but it is still a good example to keep in mind.
The function defined by $$ f(y,z) = \begin{cases} 0,&y\geq0\\ z,&y<0 \end{cases} $$ is continuous from the right in $y$ for any fixed $z$ and vice versa and continuous at the origin, but not continuous in any neighborhood of the origin.
You can make the previous example non-decreasing in both variables: $$ f(y,z) = \begin{cases} 0,&y\geq0\text{ or }z\geq0\\ z,&y<0\text{ and }z<0 \end{cases} $$