The AJATT method is actually quite a bit more in-depth than 'sit and watch cartoons'. It -does- involve studying kanji, and studying vocabulary in the form of full written sentences. Watching Japanese shows instead of English shows isn't artificial immersion - getting the artificial immersion involves everything - change your computer and cell phone to Japanese, make a playlist of all Japanese audio, etc, etc. Use earbuds rather than headphones so you still have Japanese in one ear even when you have to interact in English.
That said, I don't think it's a very practical method. Krashen's input hypothesis is the argument behind using immersion, but the input hypothesis talks about -comprehensible- input, which is estimated at knowing 90% or more of the vocabulary in what you're reading or listening to in order to even start picking up things from context ... and learning is drastically more successful at 95%+ of the content already known when learning from context.
Many success stories from AJATT are people who "didn't learn anything from classes" and then did AJATT... which means they actually -had- been exposed to basic grammar first. A lot of people also read Tae Kim's Guide to Japanese while doing AJATT... which again, is explicitly studying grammar.
If you really want to use a non-traditional immersive learning method, I would recommend looking up L-R (listening-reading) methods.
Mostly, however, I think you should just start with a textbook (ideally one with accompanying audio ; Genki is a popular choice.) I've certainly borrowed a few ideas from 'alternative' learning methods to improve my reading and listening comprehension, but I think these alternatives are best applied -after- getting down the basics.
There is certainly a lot that you will never learn without a ton of exposure to real use of the language, but there are also many things that are easy to understand when explained to you but very difficult to intuit from any kind of pure exposure-based learning plan. Which is why textbooks exist and are the most effective way for most people to start learning a language.