Ok, this question, as pointed out by others, makes no sense if you are talking about a rotating spherical planet. But let's assume you don't care about a rotating spherical planet and figure out what a system that would be most similar to ours would look like that didn't have a coriolis effect.

The simplest case would be a non-rotating spherical world. But that is boring because one side always faces the sun and gets so hot water boils while the other side gets so cold water freezes. This means all the water ends up as ice on the darkside and there is no motion of the atmosphere that isn't driven by the sensible heat flux and convection. The maximum in the solar forcing is at the equator where it is always "high noon." The surface temperature is largest here so that generates a large updraft where the atmosphere contacts the heated surface. That air rises and cools adiabatically (without condensation since all the water is on the darkside). The next part is whether we want to assume the atmospheric can radiate longwave IR (i.e., does it contain a lot of CO2). If it does then what happens is the rising air from the equatorial noon area forces the cooling air outward towards the dark hemisphere. The cool air radiates as it travels outward until it cools enough its density increases and it sinks back to the surface, where it then turns towards the equatorial noon area to rise again in the middle. In effect, a large, hemispherical toroidal Hadley cell exists, except it is driven by a sensible heat flux, not latent heat and rather than operating as a ring around the equator it operates in a ring centered around the maximum in the solar forcing. A secondary flow would exist in the out ring between the subsidence region and the darkside although the dynamics of that would be complicated since the upwelling flow would be counter to the downwelling flow from the main circulation.

If you don't let the atmosphere radiate then a single cell sets up since the air doesn't cool until it hits the darkside, at which point it sinks and flows back to the center. (Interestingly, a variant of this flow pattern exists on Mars during one of the seasons (summer maybe? I forget, but it's when Mars points its north pole at the sun).)

That example is somewhat boring, in my opinion, because there's no water vapor. A more interesting example comes from science fiction, or a variant of it, in the form of Ringworld by Larry Niven. There, the "planet" is a huge plate ribbon in space with some shields to create night and day. In that case the solar flux is uniform over the surface and there is no coriolis (well, ok, there's a little since there is some curvature to the ribbon in the longitudinal direction, but that is tiny compared to the coriolis forcing on a spinning sphere the size of the Earth) since the radial velocity of the ribbon is zero (night/day is caused by motion of the shields across the sun). Niven generally gets things pretty right, but he completely punted on what would happen to weather on Ringworld. It would be characterized by intense deep convection, tropical conditions everywhere, and no circulation to speak of. It would be like living in Samoa, all the time, or somewhere right on the equator, except there would be no trade wind edges as the ITCZ wandered around with the precession of the seasons. It would be unpleasant.

There is a third way to get small coriolis forcing and that would be a cylinder world, where the planet would be a cylinder with water and continents on the outside. This would approximate Taylor-Couette flow with a zero-shear outer surface. In these cases the flow can be extremely complicated, and likely would be since the Reynolds number would be huge. But there wouldn't be nearly as much meridional mixing as in a rotating sphere.