We don't know.
I've searched through all the books starting with Abaddon's Gate (which is when the humans first travel to the Ring) and none of them depict an event in which a ship or probe is sent through the "back" of the Ring.
The books describe the Ring as
an artificially sustained Einstein-Rosen bridge. You go through the Ring, you don’t come out the other side here.
Abaddon's Gate, p. 137
An Einstein-Rosen bridge is a type of wormhole.
I would guess that an object passing through the Ring from the back (i.e. traveling toward the Sun) would either (a) be destroyed as soon as it flew between the interior of the Ring or (b) would pass through it harmlessly and without effect (i.e. it would not enter the Ring space and would continue traveling toward the Sun in local space).
I don't think such a ship would enter the Ring space because we know that the Ring is directional when passing from the Ring space into a local solar system. For one, there is no way to pass through the "back" of a ring gate from the Ring space -- the "space" between the Rings on the boundary of the Ring space is a void (light blue in the TV show, total blackness in the books) which ships cannot enter. Furthermore, the Ring Station has a direct line of sight to each Ring's local star (e.g. the Sun) which makes it possible for the Ring Station to
blow up the Ring's local star and "autoclave" the whole system.
The Ring must be directional when traveling from the Ring space to the local solar system for this to be possible. If the Ring is directional when traveling from the Ring space to the local solar system, therefore, it stands to reason that it is most likely directional when traveling from the local solar system to the Ring space. We can't rule it out for sure, though.
I suspect the reason no one ever tried to enter the Ring from the "back" is that there is almost nothing in the solar system beyond the Ring. The Ring is just beyond the orbit of Uranus, and the only human colony beyond that is on Triton, one of Neptune's moons. Even then, a ship traveling from Neptune toward the Sun wouldn't necessarily pass by the Ring in that direction -- if Neptune and the Ring are on opposite sides of the solar system then the Ring would actually be even farther from Neptune than any other human colony in the solar system.
Given the Ring network's purpose as a "road" network between solar systems it makes perfect sense to place a Ring at the outer limits of a solar system and allow travel through it only to and from the side facing the local star -- that's the only direction ships will ever realistically want to travel into and out of. Allowing travel through the Ring in only that one direction is not really a limitation.