Easy job. If the cable runs under your hb tape, then the bar has to be unwrapped and retaped, which would be the hardest part (but you already know how to do that). To do this yourself, disconnect the brake wire from the caliper. Pull the wire completely out of the housing. Remove the housing from the bike. Cut new piece of housing somewhat longer - long enough to eliminate problem but keep the loops and bend as short as possible to avoid sharp turns in cable that add friction. You'll probably need a new piece of wire also as the old will be too short for the new housing, so figure out how to remove the wire from the brake lever. There is a lug of metal permanently affixed to the end, and usually you push the wire an inch into the lever, then move the wire sideways out of the cable-end holder. A new cable may come with two different lugs on the ends - figure out which one you need and cut off the other. Insert new wire lug into brake lever. Sometimes there is a little metal sleeve between the cable housing and lever or caliper, so slide that onto the cable first. Position the housing where it belongs, then thread the wire through it, greasing the cable as you slide it in (maybe there are some new cables that don't require grease, but I would grease them anyway). Put the cable end through the anchor bolt on the brake. Now use a 3rd hand or find some other way (toe strap, string, human hand) to hold the brake calipers firmly against the rim. Screw adjusting barrel of brake, if it has one, all the way down. Grab cable end with pliers. Pull on the pliers while squeezing brake lever - cable should run smoothly and housing should sit firmly on brake lever and caliper stop. Tighten anchor bolt, release brake, and test brake adjustment by squeezing lever. If too much space between pads and rim, use 3rd hand again and readjust cable at anchor bolt. If too little space, open anchor bolt and let 1/8" of cable slide out. When adjustment is correct, tighten anchor bolt very firmly. Trim off excess wire. Pat yourself on the back.
Oil is good, grease is better.
2007 Peter Mooney w/S&S couplers/Terry Butterfly
1993 Bridgestone MB-3/Avocet O2 Air 40W
1980 Columbus Frame with 1970 Campy parts
1954 Raleigh 3-speed/Brooks B72