Kapur, whose even par 72 in second round helped him get into the weekend rounds, is now three-under 213 and rose from tied 61st to tied 44th. In the third round Kapur birdied the first and 10th and had pars on all remaining holes.
Meanwhile the remaining three Indians missed the cut, as SSP Chawrasia (73-73), Gaganjeet Bhullar (72-73) and Jeev Milkha Singh (76-70) exited early.
Rising star Matthew Fitzpatrick holed a 15ft birdie putt on his final hole for a share of the lead at the halfway mark. The Englishman, who turns 21 next Tuesday, also sank a 30ft eagle putt on the 12th hole in his five under par second round of 67 to move to 11 under par for the tournament, a total which was match by Sweden's Pelle Edberg, who also birdied his final hole for a round of 67.