How would I remove the right hand bottom bracket cup?

No, I mean what tools would I use?

Kind of tough, eh?

Well, first you need to determine the threading. It is usually stamped on the outside of the cup. If it is 1.370 or1.375 x 24 then it is BSC. If not, based on my knowledge of your fleet, it could be Italian, French, Swiss, or Whitworth (BSA). There was even some British iron that had pressed in bottom bracket cups on the right side.

If BSC, simply take the biggest adjustable spanner you can locate and turn CLOCKWISE. This is usually the case with Italian as well. Add an extra long handle for leverage if necessary.

If French, Swiss, or BSA it is a bit tougher. The cup has threads that require a COUNTERCLOCKWISE turn- with a caveat. Since this direction tends to loosen as one rides the bike (yes... it turns backwards because of bearing rotation) they have a couple of other issues that were (often but not always) used to lock the cup in place Read on....

1) There may have been a pin or set screw placed through the bottom bracket and into the cup. This is most common on BSA, particularly on <gulp> folding bikes. This has to be located and removed before proceeding;

2) The cup may have been "sweated" on with solder. This is the worst case. If the edge of the cup is painted, this is what you are facing. Usually found on Swiss cups and sometimes on French. Carefully heat the center of the cup- where the spindle goes through- with a propane torch until it gets hot- but not red hot. Immediately douse it with COLD water and follow up with the spanner. Fair warning, I have cracked several cups this way so be sure to have a replacement;

3) Depending on the date of the bike, it might have been secured with thread locking compound. The easiest way to break this is by heating (above) or lots of leverage. Place a bolt through the spindle hole and place a nut on the other side. Place the spanner on the cup and another wrench- vise grips- on the stud sticking out... don't grab the nut. Hit the vise grips with a hammer while placing force on the spanner.

If you have a cup that has no flats but instead is splined, I suggest that you obtain the correct spanner that surrounds the entire cup and engages every spline. These can be hard to find. Failing that, get a spanner that has a single tooth, but it won't be nearly as effective.

If you just want to rebuild the bottom bracket, I suggest cleaning the thing as it remains in the frame.

You need to turn the cup in the opposite direction - to loosen is to tighten!!