Kapur will leave Muirfield with his head held high after a dramatic week where he led briefly in the first round before settling for a 68.
He was still in the mix after a second day 77 but struggled in the penultimate round when he returned with an 83.
His opening day heroics, where he shot six birdies in his opening seven holes, earned plaudits from the elite field including two-time US PGA Tour winner Jason Dufner, who tweeted, "-6 on the front nine last group off, u got to be shivving me!!?!"
The Indian, who is playing in The Open for the second time since 2006, restored some pride when he made a stunning 15-feet birdie putt on the last to end his week at Muirfield on 15-over-par 299.
"I started on such a high with the adrenaline rush. Yesterday was a bit of a disappointment when I got off to a rough start but I finished with a little bit of pride and my head held high today.