So after reading the answer to this question, I did some searching and came across this website, which said, "A very common phrase is "yoko ni naru". (Become horizontal = lay down)." So I was wondering if "tate ni naru" which would seem to be (become vertical = stand up) was used as well.
There's already a word for getting up, namely 起きる. I think that 横になる is mostly used to distinguish laying down from sleeping. Getting up is 起きる (and waking up is 目が覚める). In any case, 縦になる is not used.
