You need to keep it cool, less than +4 C, less than +40 F, it is still winter, and will be winter for 3 or more months. Unless you have a greenhouse, to keep it growing in full sun, don't warm it up until it is safe to go outside, after last frost date. (by me, it is May 1). Closed buds that are not expanding are very cold hardy. Buds loose cold tolerance as they expand. Once color shows a little, only a degree or two below freezing could kill them. Once the Ume starts to bloom, it needs to be kept above freezing to see all the flowers. But cool enough that you don't get vegetative growth started. Hence the below +4 C. The fact that it is buds further out that are browning has me confused as whether it is temperature or something else.
Too cold? Might have caused the loss of a couple buds, as would too warm. or too dry. Or too wet.
Not being able to see the tree in person, I could not diagnose from a photo, you will have to use your own observations.
Air movement. The color almost looks like bacterial or fungal issue, though not very sever, like it is just getting started. If by the time you read this it looks a lot worse, consider fungal or bacterial issue.
Do you have air movement in your winter storage area? In my well house, where I keep mine, I have a small inexpensive 7 inch diameter fan running 24 hours a day, seven days a week. It points at the ceiling, air throughout the whole cellar is moving a little. This prevented many fungal issues. First year I lost many trees until I figured out that air movement was needed.
If air movement is the problem, just adding a fan might stop the issue, if it is one of the more opportunistic fungi rather than one of the catastrophic disease fungi. If adding a fan does not stop the progression, a fungicide / bactericide might be in order. Which one? Choosing would require a better diagnosis and more photos or a visit to the plant pathologist at your local university. (In USA all Land Grant Universities have an Agricultural Extension Service that serves all citizens of the state, focused on farming).
Sorry - I can not just give you an answer. You need to parse through and choose your course of action.
