I would skip airlayering and concentrate on the main objective--the rest of the tree. The variegation you noticed is probably not variegation--Hornbeam, especially asian varieties, can produce different fall colors on the same leaf. This multicoloration can be strikingly regular on dozens of leaves, and can begin showing as early as mid-August.
In any event, if this shoot is 1/4 inch in diameter, it's not worth the trouble. True variegated cultivars aren't all that uncommon. They're useless for bonsai purposes--as they tend to look sickly and unnatural.
As noted, you can be very aggressive in removing (barerooting) American Hornbeam. Not so with Korean. I've barerooted and drastically (95 percent) rootpruned American H. at collection. If this is an American (Carpinus Carolineana) and it's been in a container more than a year, I'd have no problem washing all the existing soil off of it and planting it in completely new soil, with a root reduction--if necessary.
Root reduction and replanting set you back a year or more in developing top growth. I'd make absolutely sure the plant really needs to be repotted before doing it. I've overworked stock before --and helped kill off a very nice Korean Hornbeam by repotting on a set schedule --every two years, instead of monitoring actual root growth. I inadvertently tired the plant out with too frequent repottings...