A simple question, but unfortunately no simple answer.
You can make GBA style mugs / pixel art in MS Paint. Different popular tools are Asesprite, Gimp, Paint.net, GraphicsGale, etc…
Most people begin making new mugs by splicing vanilla mugs together, and learn how things go together and work as they go. Alternatively you can learn art foundations and then learn how to apply those foundations to pixel art.
As far as methods: anything useful for pixelart will be useful for GBA style mugs, and there’s an old collection of tutorials here: Spritans Pin Dump and here: https://forums.serenesforest.net/index.php?/topic/52389-spriters-resource/ .