I think you have the fixed and adjustable cups confused. The fixed only requires some sort of wrench, adjustable or otherwise (I realize this is called a spanner by brits and certain tool geeks, but he was talking about hammer and screwdriver as a substitute, and that wouldn't get you far on a fixed cup). The adjustable side requires a lock ring tool, set of channel locks, large set of vise grips, or whatever to get the lock ring off if you don't mind messing it up a little. Hammer and screwdriver can work in a pinch, but penetrating oil, 30 minutes, and a set of channel locks should do it. You don't need the pin spanner (which is what "spanner" usuall means on a bike) or anything for the adjustable cup once the lock ring is off, it should come out pretty easily, and if it's stiff you can give it the penetrating oil and channel lock/vise grip treatment as well. Extreme cases can require other methods, but that's not worth worrying about for now.
Don't forget stuff for cleaning out the BB shell, there will be plenty of guano in there! You should also have the BB shell chased and faced while you're in there if it's a bike you care about, and it's not a bad idea to use a torque wrench when installing the new BB.