You could simply copy your cell and then paste the formula only. That will preserve any borders.
Copy cell (CTRL+C)
Select the target-area (the area, which
you'd normally drag that cell down to)
Right Click the selection
-> Paste special -> Paste Formula Only
I figured out a way that is almost as good. Early on in Google sheets double clicking the paintbrush icon worked. Why Google removed that I don't know. But while you can't make the paintbrush icon "stick" anymore, what you can do is:
click the cell (or range of cells) whose format you want to copy.
click the paint-format paintbrush icon (to copy format).
For now, I have worked around the issue by rotating the text down instead of up. Although this doesn't achieve the alignment aesthetic I wanted, it will suffice for now, and clipping end of value text works properly.
You have a long header cell that you have rotated to save space. However, when the height of the row is shortened, a "long" value is effectively right-aligned so that the end of the value is shown rather than the beginning.
As noted by @marikamitsos, it's not possible to do what you wish.
However, wrapping the text is a good work-around to display the ...