The characters in 昨日 mean "previous" and "day," so it is a standard combination for "yesterday." As for the pronunciation, Japan has a lot of words that don't appear to follow "the rules." (In this case, 昨日 can also be pronounced "sakujitsu," which is closer to what you might expect.) It probably stems from Japanese being first a spoken language onto which the kanji were later adapted from similar Chinese written words.
I have seen some pronunciation transformations that are completely unrelated to the typical sounds of the characters, sometimes even applying English words. For example, 物語 is normally pronounced "monogatari," and means "tale," as in "fairy tale." But there is one song I know where the lyricist applied the pronunciation "ストーリー" to these characters, from the English words "story."