I don't know if you mean Alfred's piano books, but those are the ones that I use with my students. If you're not using Alfred's you might have a look. There are some sample PDF pages.
Mikrokosmos is a great series, if you like Bartok. Getting students to play the pieces if they don't like Bartok is more trouble than it's worth. Fingering is a flexible concept; the fingerings in books are suggestions, not rules. (I have five different editions of the Beethoven sonatas, and the fingerings are often very different in each. Beethoven had a number of fingering suggestions of his own in his manuscripts, and the editors often happily added their own suggestions as well in the same spots.)
The best thing to remember about fingering is that whatever lets you play the music the way you want to play it is the right fingering. On the other hand, there are general fingerings that work for typical passages such as scales, and it would help you to learn these.
A few general rules: avoid putting the thumb on black keys in stepwise passages, especially passing the thumb under a finger to a black key when the finger is on a white key. Learn to play a note with your thumb (either black or white key), and, while holding it down, play the note an octave higher with your eyes closed. Use both 5 and 4, and if 3 is comfortable, learn that too. Do the same with 6ths and 7ths. Plan ahead for stepwise passages in such a way that you avoid "running out of fingers". In other words, work out ahead of time which notes you will take with your thumb.
With all that said, in your question, if your fingers are in the middle C five-finger position, you could just reach your thumb over one note to the left to play B3. If you have to play other lower notes, it's probably best to pass one of your fingers over the thumb and play it that way. So, as you can see, what fingering you use depends on what else you have to play. Just avoid "running out of fingers" by planning ahead.
Now, scales will teach you a lot about fingerings that work and fingerings that don't. Since you're already musically advanced, I suggest that you begin working on scales in two octaves. Most people start with C major, because it has no sharps or flats to worry about, but it's among the most challenging of scales to play evenly as well. I often recommend E major as a starting scale, because the fingering is the same as C (and most of the other scales) and you have plenty of black keys as guideposts to help you out. I'm also a fan of the Hanon exercises, at least the earlier ones in the first book. Hanon suffers from Victorian absolutism, but if you take his pronouncements about his exercises with a large grain of salt they are very useful to teach you things about how the hands work in realizing various musical patterns. The second Hanon book has all the scales and arpeggios with appropriate fingerings so you might find it useful as well as a reference.
So, by all means, work with Mikrokosmos if you like the music. It's great music, even the simple stuff at the beginning. Don't get too caught up in "correct" fingering, but also don't learn bad habits either. Bad habits are habits that prevent you from playing the music the way you want to play it.
Finally, your one year goal is entirely realistic. Just pick up music that interests you and see if you can play it. If not, set it aside to pick up later. Keep practicing, and avoid mistakes. If you are making mistakes, slow the music down and break it up. If you're a pro, then you know the drill.