1+1D curved spacetime diagram example

This is a very basic question about General Relativity. I haven’t take any GR course yet.

Suppose a flat spacetime with one space direction and one time direction, as follows:

Now add a mass at rest at $x_0$. The world line of this mass can be drawn in the spacetime diagram as follows (red line):

Now I wonder how the constant time and constant space lines would look like?, i.e. I would like to depict the grid of this spacetime.

My reasoning (may be wrong) was as follows: Over the $t$ axis I marked down the units of time. Then, I know that inside a gravitational field clocks run slower, so I marked down (over the mass’ world line) the slower units of time. Like this:

What I did next was just to join these marks as follows:

I don’t care about the exact shape of the constant time lines, I just want to know if this reasoning is good.

I also wander how are the constant space lines… If I consider the following

then a very small test mass placed at rest at some $x$ near the big mass will eventually feel an attraction and at $ttoinfty$ will collide with the big mass. This must be wrong because in real life masses do not need infinite time to fall. But I cannot manage to find the constant $x$ lines.