Ancient Pictographic (before Paleo) Hebrew is like this too. Except they only have 22 letters, there's a 23rd but it is no longer part of the 'Aleph Bet'. Hebrew originally had no vowels, and thus today they only put the vowel system in when it's a word people don't normally know how to pronounce. Chinese is similar as well, and I'm sure there's a couple more languages.
Far from being like this, the only major kind of writing that Japanese doesn't use is an abjad. It uses logograms (kanji, like what this article shows), two different kinds of syllabary (hiragana and katakana), and an alphabet (romaji - the Latin alphabet).