Hue-shifting is just changing the hue of shadows and highlights relative to the base colour. If you do this consistently (e.g. shifting all shadows progressively towards some hue, and shifting all highlights towards some other hue), it has the effect of unifying the colours.
Another way to make disparate colours work together is desaturating them somewhat, especially the more dominant ones. I did an edit with both changes:
Your "yellow" was very desaturated and green, so I made it more golden-looking (more orange, really).
I made the light blue a bit more greenish and less saturated, and the blue shadows more purple and more saturated. Combined, this still gives the impression of dark blue, but without being so boring. Purplish blue is also the complementary colour of yellow-orange, so those colours work better together than blue and greenish-yellow. If you pay close attention to the design drawing of the character, you might notice they also used golden yellow and purplish blue instead of greenish yellow and pure blue.
Try to avoid using pure grey. Light is rarely completely neutral, so grey objects usually have some colour to them. I made the sleeves and trousers bluish with purplish outlines.
I changed the background colour to grey, because that makes it easier to see the colours as they actually are. Don't use saturated backgrounds to work on, as they throw the colour balance way off. The one exception where working on some bright colour is a good idea is when you know the sprite will be seen primarily against a background of that colour, so your colours need to work with it.
Colour theory is a vast topic and I doubt I've explained things well, but I hope this helps anyway ;~;