If you're loathed to buy a new tool here's a couple of thoughts - I'm not advocating them - it's up to you.... (but I've been successful with both in the past but they are risky and painful) - if you've reasonable craft skills and really don't want to buy a tool, they might be worth considering......
1) Hammer & punch
The photo above suggests someone's had a go at this cup with a centre punch anyway - so you've nothing to loose. you basically need to rotate the cup by locating the punch in the hole at an angle so that hitting it drives it CCW as well as in - too much of an angle and the punch flies out (ouch) not enough angle and there'll be no CCW cup rotation.
2) Angle grinder & Chisel (brutal)
Use a wrench to remove the fixed cup from the other side and take the axle out.
Use an angle grinder to cut a slot in the protruding cup (careful not to cut into the BB shell) . Now you can use a nice big cold chisel as a screwdriver by clamping it in a vice and rotating the frame. (or just spin it out with a chisel instead of the centre-punch)
If you're loathed to buy a new tool here's a couple of thoughts - I'm not advocating them - it's up to you.... (but I've been successful with both in the past but they are risky and painful) - if you've reasonable craft skills and really don't want to buy a tool, they might be worth considering......
1) Hammer & punch
The photo above suggests someone's had a go at this cup with a centre punch anyway - so you've nothing to loose. you basically need to rotate the cup by locating the punch in the hole at an angle so that hitting it drives it CCW as well as in - too much of an angle and the punch flies out (ouch) not enough angle and there'll be no CCW cup rotation.
2) Angle grinder & Chisel (brutal)
Use a wrench to remove the fixed cup from the other side and take the axle out.
Use an angle grinder to cut a slot in the protruding cup (careful not to cut into the BB shell) . Now you can use a nice big cold chisel as a screwdriver by clamping it in a vice and rotating the frame. (or just spin it out with a chisel instead of the centre-punch)