If the Universe did not expand of contract, the picture would be a cylinder - that is correct. If the expansion was a constant, it would be a cone ... providing the singularity at the point that we talked about getting rid of earlier.
To avoid the point, the geometry requires the time axis to curl around so the space-time picture resembles a hemisphere. In the picture, there are a few extra tweaks to account for quantum effects in the very early universe - the hemisphere looks like a nipple in the bit labelled "quantum fluctuations".
In the other direction, there is a trumpet-like flaring ... this is due to the currently observed acceleration in the expansion.
The artist has drawn that end open because any ideas about how the Universe ends up is pure speculation at this stage.
I have a post in that thread talking about the bits I see as misleading in that picture.
Anything we do like this can only be an analogy - and all analogies have limitations.