How do I make a character, who's kind of shy, show his emotions?

Writing really is the study of human beings and how they express 
themselves. Every opportunity you can - EXAMINE (like a being from 
another world) how the humans that surround you express the vast array 
of emotions that color our daily lives.

You’re right, some people are withheld - 
the merest lift of an eyebrow says volumes about what's going on in 
their heads. Other are like fireworks displays - about everything; 
love, hate, happiness, boredom it all gets colored with some 
incredible display of emotion. The films that affect us the most are 
the films where the writer has created a VISUAL language that relays 
even the tiniest detail of a characters INNER landscape (emotional 
landscape) to us, the audience.

So, give some thought to the multitude of NON VERBAL bits of 
communication that are peppered throughout your day... your child 
opens it's arms wide for a hug when you come home; but doesn't SAY 
anything. Your boss merely crooks a finger in your direction and 
disappears into her office. You know you've been summoned. At a chai 
shop, a handsome fellow merely meets your eyes over the lip of his 
cup, which mostly hides his smile of acknowledgment.... you TELL ME 
what that means!!

Start cataloging in your mind the various ways people SHOW 
us every day what they're thinking... instead of using words. Do this for a 
month and you will be on your way to understanding one of the key 
elements of screenwriting. Show - don't TELL what the characters are 
thinking.

I hope this helps. And I do wish you luck with your screenplay . Let me know when you’ve completed your script. I’d love to read your work!