Can I just add air to an air-over-water pressure tank?
If you have a fitting you can connect a compressor to, you don't even need to turn the pump off - pressure from the compressor should keep the pump from turning on.

I'd strongly suggest, when funds and time permit, some replumbing to put in valves you lack and considering a bladder-type tank, but for cheap and quick, if somewhat of a hassle, just draw down water until you get near 40 (but not so near as to turn the pump on) and pump up to 80 with the compressor, repeat as needed or until you have the water level you want for now. You will be repeating, given a non-bladder tank and evidently no or malfunctioning (common) venturi (aka micronizer?) valve to automatically add air. So if you don't plan on changing the tank, plan on making it easy to connect the compressor. I'd use a quick-connect coupling. Do make sure the compressor you use for this is "oil-free" - your drinking water supply could get nasty with an oiled compressor adding air (and just a bit of compressor oil) to it.

The pressure switch does not need to be submerged, by the way. It senses pressure, and does not care (or know) if it's sensing air pressure or water pressure. The "standard drill" for bladder tanks is 3 psi below the cut-in pressure when empty, i.e. 37 PSI in your case (however, it's not a bladder tank). But given no place to flush from (other than shut all water uses off, pump to shutoff, then flush from outside hose connection, which will help with the tank debris, but not anything in the pipes to the softener) and no basic filter ahead of the softener, adding air without fully draining is probably safer, since it will stir up less debris.