Here’s the scoop on your Neon’s manual. You are entirely correct. The transmission does not have a filter, as opposed to an automatic that does have a filter (it’s due to the fact that the automatic requires lots of fluid to make it work).
As to pulling the tranny pan, it is also relatively easy. After draining the transmission fluid, get under the car (or put it up on a lift if you have access to one; a set of ramps or a hydraulic floor jack will do as well, though you have to remember to block the wheels as a safeguard against rolling) and carefully unbolt the transmission pan.
Carefully remove the pan and put aside in a pail or something similar to prevent dripping on the floor. Let it dry and then check the edges to be sure that there are no dents or other issues that would prevent a tight fit when you close it up. Do the same on the transmission housing, as well. Be sure to obtain the right gasketing and, following directions on the gasket box, line the edge of the transmission pan with the gasketing material and place it carefully on the transmission housing. Then, carefully torque down the bolts (14), being sure not to overtighten them as you will distort the housing the problem may just continue. It’s a good idea to tighten the bolts with a torque wrench. Each of the bolts has a certain metric to which it must be tightened and no more. You can find them in a shop manual for your Neon. If you do overtighten the bolts, then, as noted, you might damage the rim and the problem with leaking may just continue, even though you solved it with the installation of new gasketing.
When everything is buttoned up correctly, refill the transmission with 2.6 liters of ATF-4 and you are good to go.