Kapur, lying eighth overnight, shot 69 and moved to nine-under but slipped three places down to tied 11th.
At five shots behind the leader on a crowded leader-board, Kapur can still make the charge on the final day.
Meanwhile, the other Indians had a bit of struggle as Jeev Milkha Singh (74) dropped to tied 48th, Chiragh Kumar (71) was also tied 48th and SSP Chowrasia (76) was way down in 71st place.
Local star Masamichi Uehira fired a third round score of six-under-par 65 to grab a one-shot lead over Filipino Juvic Pagunsan after 54 holes.
Uehira is now 14-under-par 199 but with 14 players within five shots at the 150 million Yen (approximately USD 1.9 million) event sanctioned by the Asian Tour and Japan Golf Tour, the final day will see some exciting finishes.
Kapur said he had managed some rest after jet-lag and the travel from Europe but could not get going on the front nine.
"I was a couple of over-par after nine. I really fought back hard. I had a good rest last night. I think my first fairway of the day was on the ninth hole. I was completely out of rhythm," said Kapur.
"I dug deep and I told my caddy, 'let's shoot a 31 on the back nine and get back into this.' I almost did that if I birdied the last but you can