First, the Earth and the ship will never leave the observable universe of the other. That is not possible. What you mean to ask is whether the ship can cross the *cosmological event horizon* for Earth.
This event horizon is at the farthest possible distance that a light signal emitted *right now* will eventually reach us. Any light signal emitted farther out will never reach us. This event horizon is also shrinking over time. Eventually, the cosmic event horizon will contain only our Local Group of galaxies.
) shows what you are looking for. The blue line segment indicates the diameter of the cosmic event horizon as it is right now; it is currently about 16 Gly away. (Note how the event horizon diameter shrinks over time.) On this diagram, the path of a light ray is at a 45-degree angle to the horizontal parallel axes. So suppose we send out a light signal right now, and then when it reaches the event horizon it is sent back (idk, by some bored alien or whatever). What does the path of those two light signals look like? Like the two green line segments [in this image](https://imgur.com/BAufBHt
So how far out does that light signal travel? From the axes, it looks to be about 8 Gly. But that's in co-moving distance. So that means the light signal gets to a galaxy that is currently 8 Gly away from us. By the time the light signal gets there, space would have expanded and the light signal would actually be a proper distance away of greater than 8 Gly. How much has the universe expanded by then? The vertical axis on the right gives the *scale factor* which is the ratio of proper distances at that time to proper distances today. If we trace a horizontal line from the end of the first line green line segment to the vertical axis on the right, we see that the scale factor is about *a* = 2.1. So the light signal will be about 8\*2.1 = 16.8 Gly away from us when it reaches the event horizon. [This calculator](http://home.fnal.gov/~gnedin/cc/
) seems to say that a scale factor of 2.1 should correspond to an event horizon proper distance of 17.1 Gly, which is about close enough. (These are tough calculations and the calculator and author of the graph may be using slightly different approximations. I am also estimating distances using the axes in the graph, and so I can just be way off.
How long does it take for the light signal to get there? Ehhhh, that's a tough calculation and not one you can read off from this chart since the time axis is in units of *conformal time*, not *cosmological time*, which is how we usually think about time. If you trust that the scale factor at the time the light reflects back is about a = 2.1 and you trust [this calculator](http://home.fnal.gov/~gnedin/cc/
), then the light signal reaches the event horizon after about 12 Gyr (that's 12 billion years) after it was emitted from Earth. Strictly speaking it takes an infinite amount of time to come back if it is reflected right from the horizon exactly. So if the light is reflected just before the horizon, then the time until it comes back can be arbitrarily long by just reflecting the signal at distances arbitrarily close to the horizon.
So what about your spaceship? Well, the spaceship has to travel slower than light speed. In the graphs I have provided, the path of the ship is not exactly a straight line, even if it travels at constant speed. But the slope of its path must always be greater than 1. Which means the path of the spaceship must lie within the triangle formed by the two green line segments and the x = 0 vertical line in the graphic. In any event, the distances and times gives above are upper bounds and a spaceship can reach those bounds arbitrarily close by just traveling faster.
Finally, as for the direction the ship or light signal travels, all of these calculations assume the light/ship is emitted radially outward and then is reflected (or comes back) radially inward. That is, the paths are directly out and directly back.