With brifters, the easiest way I have found to replace brake cables (not housings), is to unscrew the nut holding the brifter in place on the bar, insert the cable through the brifter, into the housing, and push it in until it can be pulled out the other end of the housing, then remount the brifter. Has saved me from re-wrapping tape.

This is after having pulled back the hood covering the lower part of the brifter.