Can one build up the worn section via welding?
Usually, Yes. But not always.
But is it economically viable to do so? Welding time, welding materials, lathe machining time after welding all add up. What labor rate and overhead rate are you running?
For $40 to replace the entire shaft, my really rough estimate would be that you would have to do the ENTIRE weld-and-remachine job in less than a half hour elapsed for it to even be close to economically a 'good' choice to do that task.
That half hour includes all the welding set-up time and all the lather set-up time.
psst, do not weld round-n-round if doing a shaft build-up repair. Run a weld bead length-wise, then rotate the shaft and run another weld bead length-wise, repeat until the shaft is built up as desired. Don't turn the shaft into a banana, or you then have to add in more time and effort to try and straighten out the now bent shaft.
Also watch out that some shafting is made with 'free machining' steel (either leaded or has sulfur or phosphorous in it). Those materials are NOT weldable.