I've got parts that could be combined to do the job but I have not actually done
this type of measurement (yet).
https://www.satellites.co.uk/forums/attachments/arduino_stepper-jpg.72933/
...shows a component to get the angular resolution and precision needed.
Manual operation gives 0,01 degree but microstepping the motor gives 4,5 arcsec
in 1/8-step mode when driving it electrically. The other part needed is the
absolute reference to what zero degrees is. Spirit-levels used in surveyors equipment
can be found with different resolutions, I think I've got one that is quite sensitive
with a millimeter resolution over tens of meters in range. Spirit levels can be adjusted
(zeroed) accurately without other tools than lots of patience.
A stable metal profile with a rotational stage with a spirit level on the moving part
would do the job. (The rotational stage in the picture is a Standa 8MR180
http://www.standa.lt)
A simple but accurate zero reference plus complicated mechanics in principle.
Electrical sensors with 0,01 degrees of resolution are horribly expensive but they do exist.
edit - sprellink erors