I have built several h22's. In that time not one leaked any oil. However from the Honda tech's I have heard that some do leak a little bit of oil. I think 1 complete quart every 1k miles is still slightly excessive. You may want to consider compression check and a leak down just to be thorough.
Honda's offical policy used to be anything under 1 quart every 1000 miles is acceptable and they wont warrenty it. I argued many times with their engineers that it is excessive but that is what their policy used to be up till i opened my own shop in 1991. after that i dont really know what their policy is. I personaly dont like any oil consumption but some motors do "use"oil.
All H22a motors burn oil. All of them. Don't bother wasting money using synthetic in them. The VTEC oil pressure actuated system makes it happen in combination with the H22a's crappy piston rings. Check your oil at every fill-up that's what I do on mine.
I know a few people that got warranty short block replacements..but they were over 1L per 1500 kms..
The only synth I didn't find as much burning ws German Castrol 0w30 but that was on a built motor with iron sleeves and Endyn/Wiseco pistons and rings.
It's not a problem persay, but it's standard H22a behaviour.