Meizi (or ume, as they're known in Japan) are green apricots, although they're not simply the unripe fruit that will, if left on the tree, ripen into the beautiful golden-orange variety that we eat fresh later in the year (or eat dried). Green apricots, which are very tart, are rarely eaten as they are. More often, they're infused into alcohol to make a variety of drinks; dried with salt and/or sugar and/or other flavourings for intensely flavoured tart-salty-sweet snacks; or cooked into preserves. It's not the easiest fruit to find because its season is brief (start looking for them now). And because so few people know what to do with them, not many vendors carry them. When buying green apricots, look for firm, evenly coloured pieces without dark spots, nicks or bruises.
The easiest thing to do with green apricots is infuse them, although it will be months before you can taste the results. Pull off and discard the tiny stem (it's not easy, as it's very tightly attached), then pierce the fruit in several places with a sharp, sterilised pin. Weigh the green apricots and take note of the amount, then put the fruit in a sterilised glass jar. Add clear alcohol such as sake, shochu or vodka to the jar so the fruit is completely submerged. Place in a dark spot for several months, shaking the jar every once in a while. The fruit will turn khaki green. After three months, add sugar - half the amount (by weight) of the fruit. Seal the jar again and leave at room temperature, shaking every day, until the sugar is dissolved. You can leave it for a couple more months before refrigerating and drinking; the fruit can be eaten at this point, although it will be very strongly flavoured with the alcohol you used.
For preserved green apricots, de-stem and pierce the fruit as above, put it in sterilised jars, then cover it with a balanced mixture of granulated sugar, salt and spices, such as star anise, licorice root, cinnamon sticks (broken into pieces) or dried tangerine peel. The sugar and salt will draw the liquid out of the fruit and the flavourings will be drawn in (this takes a long time). Shake the jar frequently, and leave for several months before eating. The preserved fruit keeps for years, as long as it's stored correctly.