as he is finding it difficult to engage with others and is spending a lot of time cooped up in his room
What is the nature of his difficulty in engaging?
Is he going to seminars, tutorials, lectures? Can he make superficial social chit chat before or after a lecture or seminar.
Does he have an interest which he could explore via Student union clubs & societies?
Could he go to the gym, for example, just to get out of his room? Or go for a walk each day.
He needs to step over the threshold, and try to make a physical & mental first step. Just a simple first step - but he needs to do it.
Did he have difficulties at school? Is it because in the past, his social interactions were organised for him (by you, or his school?).
The self-help materials my university provides for students are excellent, and he should work through what they provide.
But ultimately, he'll need to find strategies to get over his shyness. It helped me (I was very shy until I was about 25 when I realised I just had to get over myself) to try to work out what caused me to shrink back from social interaction. For me, being fairly relentlessly bullied for a lot of my time at school was part of the problem , & getting to university & realising it was full of other girly swots who liked talking about ideas & enjoyed studying. was tremendously liberating.
The simple mental health "hygiene" advice we give students as personal tutors is to attend classes, get some exercise each day (outdoors is best if the weather allows), try to develop a regular sleep pattern, and think about a non-studies activity that gives you pleasure.
Very simple things, and part of learning to take care of yourself as an adult.