For upper right most, I’ve seen it written upper-right most, upper-right-most, and with no hyphens at all.
What makes the most sense to me is upper rightmost, but it’s hard to tell that upper right is the adjective to which most is being applied.
As John Lawler comments, and as this chart makes clear,...
...rightmost isn't normally hyphenated. And there's no need for one in upper rightmost.
However you punctuate it, this is appalling English. Why don’t you say “closest to the upper-right-hand corner”?