It only takes a few days to learn the two phonetic syllabaries, hiragana and katakana (which is probably what you mean by alphabets).
I did it simply by writing the syllabaries out in order, learning one 5-character row at a time and commiting it to memory. It took me about 5 minutes out of every hour for a week to learn both syllabaries. (Breaking it up into small sessions through the day helps test and build your memory; you can't really just sit down and memorize it all in one long sitting, unless you're blessed with a phenomenal memory or have some highly developed memorization techniques.)
If you want to be more modern about it, there are tons of free flashcard programs and memory 'games' for practicing recognition. I honestly think though that actually writing (in correct stroke order) is important, which isn't to say you can't use flashcard programs or memory games. Google around (or search these forums) for 'learning kana' or the like.
