Kapur, who shot 67 in the first round, is now seven-under 135 for 36 holes.
Kapur is three shots behind leader 46-year-old veteran Hiroyuki Fujita (68), who after four birdies and one bogey in second round is at 10-under for two rounds at the Higashi Hirono Golf Club.
Three other Indians made the cut. Jeev Milkha Singh, who shot 67 on first day, had an even-par 71 with two birdies and two bogeys to be tied 25th at four-under.
SSP Chowrasia (67) and Chiragh Kumar (72) also ensured weekend action at one-under and tied 59th for two rounds.
The cut came at one-under 141 and 72 players made it. Chowrasia rode on an amazing streak of four birdies from 12th to 15th to ensure he made the cut after a first round 74.
Missing the cut were Himmat Rai (75-69), Digvijay Singh (75-69) and Jyoti Randhawa (72-72), Anirban Lahiri (72-74) and last week's winner, Gaganjeet Bhullar (74-79).
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Kapur birdied the second, sixth, 12th and 14th and his sole mistake of the day was a bogey on the 10th. He is tied with three other Japanese players.
Kapur, who flew in from Italy on Tuesday for the tournament, said, "It is pretty good except for the fact that I haven't had much sleep in the last couple of days. I came straight from the Italian Open and I'm a bit jet lagged.
"I got two hours of sleep last night so I didn't have much energy out there and wasn't hitting the ball the way I like."
"I didn't have my 'A-Game'. It was probably a 'B' off the tees but I was sharp on the greens. Every time I seem to miss the greens, I get an up-and-down and that sort of took the pressure off the round. I think that was basically the story of the day." MORE