I kept googling "sleeve tattoos", but I can't find sleeve tattoos like the ones in the picture above with that term. Is there a keyword or a word that refers to that particular tattoo style?
Sleeve is the term used for tattoos that cover the entire arm. Tattoos that cover only part of the arm can be called a half-sleeve, although this usually means a tattoo that covers the upper arm and shoulder.
I suppose these could be called partial sleeve tattoos, or simply arm tattoos.
Given the fantasy context, an author writing about these tattoos would probably say something like:
The fairy had eldritch, glowing tattoos on her upper arms and her wrists.