Should I ask him to go out with me or not?

I think you should ask him out go for it! He's just really shy i use to be like that in high school i liked a girl and i used to stare at her dream about her and stufff but i never really got the courage to ask her out but eventually she started talking to me and we became close and started went out for two years.i think you should do the same ie. go up to him start a conversation and give signals that you like him.