Honestly, it isn't that much. There is only a single "she". I can see the repetitive "her"; but this isn't a big issue for my taste, at least not in this extract.
However, you could try this:
Clitter, clatter. The heavy metal case carrying her belongings rolled over the small divides in the polished white floor, sliding sideways with every disoriented turn Akio made towards the exit. The wind howled outside the airport, displacing a piece of dark brown hair on the woman's cheek. Goosebumps crawled amongst the stubble growing on her legs. Akio's cold hands rubbed together to create a pocket of warmth which she clung to. Her feet were glued to the ground as her mind wondered of the new adventures that lied ahead. It was a new beginning.
Basically, the advice here is, to use "she" or "her" interchangeably with the character's name, gender -or simply "the", if it's obvious who we're talking about- and to try not to have the same one repeat many times in a row.
You might feel like things are repetitive, but truth is, most books are written with what feels like two alternatives to the pronoun, and trying to add fancy ways to describe it all the time may clutter things up too much, though that's personal preference. Clean up what bothers you most, leave the rest alone, I'd say.