I've air-layered young branches successfully in May (Tennessee) and cut off the tree in October. Most of my air-layers were on Prunus mune 'Contorta'. 1" wound all the way around a branch and painted with pure rooting hormone (I was lazy) and covered with long-fiber sphagnum moss and plastic wrap. Twist ties on top and bottom then wrapped whole thing in aluminum foil. Checked for moisture about once a week then every other day during height of summer heat. I punched a few holes in the top and bottom of the plastic wrap to allow water in and out. Pure rooting homone is not advised as it caused a lot of callus tissue to form and decreased my number of success rate. Hormone may not even be necessary.
Semi-hardwood cuttings 4-6" long taken in June with 3-4 leaves and 4000 ppm KIBA (This rate was most effective for numbers of roots but no hormone also worked). Wounding the base of the cutting did not seem to increase rooting success. It did cause rooting to be "lopsided" as roots formed from the callus tissue. Media was generally. Used individual cell trays (Rootmaker) as ume have very tender roots and it's easier to transplant them later. Media 70% perlite, 30 % peat moss. Rooting ume isn't hard, but transitioning them to container culture and over-wintering year one is not easy. I used an automated mist system that turned on 3 seconds every 5 minutes the first week, 3 seconds every 10 minutes for about 6 weeks, then hand watered as needed. Mist system kicked off at 6 am and stopped at 4 pm. I rooted 'Josephine, Ushiro Chiremen, Hokkai Bungo, Mitsubara Red, Fragrant Snow, Peggy Clarke' etc. Never rooted the weeper 'Bridal Veil' with consistent results; usually a few out of 100 cuttings.
If you don't have a propagation house, try 6 inch cuttings about a month after the leaves harden in pure akadama (not high fired). Cut the leaves in half and place container in about 30-40% shade. Mist when possible. Success rate is lower but that's how I've heard it done on shoe-string.
Camellia Forest Nursery in Chapel Hill, NC (great mail-order service) has a number of varieties of ume not to mention 100's of camellias and other rare plants. Good people
