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With a good theodolite and proper procedure, it can be done. It will take two setups per column. You will need to setup perpendicular to the column. Plumb the steel column with the vertical crosshair of the theodolite. Have a person hold a wooden rule along the side of the steel column...

Are you thinking of a 4ft Smart Level? At ±0.2°, it is not accurate enough. I own a Topcon DL-502 digital level. It is extremely accurate (±0.6mm), but no way to use it for the OP's intended purpose. ~LeeGreen.com Licensed Surveyor in NY sUAS Remote Pilot

"Would you considered 1/600 between column splices?" Yes, if that is the required tolerance. That would equate to 78mm allowable deflection in the entire 47m. ~LeeGreen.com Licensed Surveyor in NY sUAS Remote Pilot

If the monuments are disturbed or destroyed they will need to reset by a licensed land surveyor. If you know they are going to be destroyed, be certain to hire the surveyor prior to construction, so he can accurately locate original monuments location.

Either this is a simple/trick question or we need more information. The angle at the PRC for all four quadrants is always 90°.

I have performed many "vertical column plumb surveys". A task that requires this type of precision must be done with a very accurate instrument and procedures. The instrument needs recent calibration. Are you prepared for this? If not, I suggest you hire a licensed land surveyor with experience...