Idiagnosed with central cervical canal stenosis and far lateral disc protrusion and a disc extrusion indenting cord. how serious is this. arm pain ?

Cervical Stenosis: This is serious and something you need to see a neurosurgeon about. Do not do any strenuous activity and be very careful of any falls as these may worsen your condition and cause paralysis. Good luck

Depends on meaning: If by SERIOUS you mean, "dangerous, deadly, or mortal" then, none of the above. But it certainly can be painful & result in more symptoms as time goes on such as weakness & sensory deficits if untreated. If in fact, an examination confirms involvement of the cord then, a more aggressive approach may be indicated which could include a recommendation for surgery if conservative means fail.