I do not keep a tank running as a hospital tank.
When you are going to put a fish in a hospital tank it is usually nowhere near a full bio load, so it does not matter if the tank is cycled or not.
Fill it with water from the main tank. Not because that water is cycled (there is not such thing as cycled water) but because that is the water the fish is used to. You could use some proportion of old water and new, and it is like a water change.
Take a little media out of the filter that has been running on that tank, and use that in the filter of the hospital tank.
Look at it this way: What % of the fish load are you removing? 1 fish, 2 fish out of 10? That is 10% or 20% of the bioload. The filter can spare 25% of the media even if you were not removing any fish, so go ahead and take 25% of the media for the hospital tank.
Set up for a hospital tank:
Bare bottom: Allows easiest vacuuming, and to see any shed parasites. Put the tank on a dark surface, or put a black towel under it. Dark is less stress for the fish.
Take clippings from any of the tanks and let them drift in the water. Gives the fish a place to hide, and a bit more bio filter. When you are done you will probably throw away these clippings.
A rock or a chunk of driftwood. Something for the fish to hide behind, but can be sterilized or thrown away.
Filter, heater, light: If the filter is too strong, and you do not have another one you can put something in the path of the water flow to divert it. If that still does not work a small air pump can run a small trickle of bubbles. Something to create a little ripple at the surface. Put the media from the cycled tank where there is good water movement. If you do not have a spare heater you can wrap the tank in a towel, and keep it in a warm place, but a heater is so important that you should go get one and keep it with this tank. Many diseases are made worse by the stress of fluctuating temperatures.
Many fish will not like the light when they are sick, so it is OK to leave it off most of the time. But the plant clippings will die, and the light contributes some warmth to the tank. I would rather leave the light on and add more clippings or floating plants to shade the fish. If you use an incandescent bulb that adds more warmth to the water.
The basic medicine cabinet would contain long lasting things like hydrogen peroxide, salt, water changes, Melafix and Pimafix, a wormer (I usually have some Prazi-Pro hanging around).
But other meds, that age fast, I would rather diagnose the problem and buy fresh meds.
About your Neon: Do some research about Neon Tetra Disease.