You're wanting at least 10V across the cell (more is better IF you can get it which basically boils down to what current you run). At low currents, with 20 drops of electrolyte in the water, it may be difficult to even reach 10V. When I was running 3ma, I had to back the electrolyte down to about 16 drops just to get the cell voltage high enough. Several others here had to do the same thing. Up over 10ma you shouldn't have a problem.
So 10 gone (or whatever your cell voltage winds up) from that wall cube immediately.
Now the limiter. If you're building an LM317 type limiter, figure you need about 3-4V for it to function (seriously). So now you're up to 13-14V. But, you don't want the limiter to be able to BARELY function. It needs some headroom (say 3-4V minimum) so now you're up to 16-18V. See where I'm headed?
I use a 19.3V laptop charger and that works well for me but if you're buying, a small, low current 24VDC wall cube is perfect for beginners.
As far as current, the limiter doesn't use any current. All current that passes through it, passes through the load so if you're running even 20ma, you need a power supply capable of 20ma (0.02amps). 1/2amp is way up in the stratosphere here, so you're good.
Be careful though with thinks like that fancy thing shown. For sure that uses current that doesn't make it through the cell to function so you need to make sure the power supply puts out enough to cover the power needed by the device plus that which you endeavor to run through the cell. I still think 1/2amp is plenty but if you choose to use other electronics, check to see what their operating current is before you buy a power source.