How do I remove this style of bottom bracket?
When I fixed up my old bike it had a typical BB with fixed and adjustable cups. I planned on replacing with a cartridge BB so I didn't want to get the special tools just for a removal operation. Though I did get the special toothed socket needed to install the new BB (a cheap version online since I'll only use it a few times). Using a slot screwdriver, or better, pin punch or cold chisel, and mallet/hammer, you should be able to get the cup turning. Make sure you're driving it in the correct direction. To remove non-drive side, drive it counter clockwise. To remove drive side, turn clockwise. A small amount of penetrating fluid can help too.

Edit:
Forgive my crude sketching abilities, hopefully this helps. As in the side view, hit the notch perpendicular, and as in the top view, keep the pin/punch as parallel as you can to the face of the cup. (in this sketch this is for removal of the non-drive side). Depending no how stuck it is, it might help if someone holds the bike upright with the brakes on.