I am a lot older than you are and went through much the same thing. Listen to what I have to say.
I was adopted as an infant, then after my real (adoptive) parents died, I sought out my birth parents and was successful.
In the process, I gained five half-sibs. One is my mother's other daughter (half-sis), four are my father's sons (half-bros). All this happened 14 years ago, so I have some perspective of time.
Of course, I am older than all of them, for obvious reasons.
My half-sis first heard about me when she was 20 (or so). Her mom spilled the beans about having had a baby out of wedlock. My four half-bros knew nothing. Their father had told his wife about his illegitimate daughter, but they never told their boys. After I contacted bio-dad, he had to make four incredibly difficult phone calls. ("Dad, have you been drinking???? Are you out of your mind???")
In terms of sibling relations, it's been a mixed bag. Half-sis has been used to being an only child, and it was an adjustment to gain a sister who was father along in life than she was. (Got college degree earlier, got married, had older kids, etc.)
Of the four half-bros, again, a mixed bag. One guy really likes me, the other has nothing to do with me (I think it's his wife's issue, but never mind) and the other two are somewhere in between. They all had to get used to being "one notch down" in the birth order. Older brother was no longer his father's oldest child, etc.
Here is what I suggest for you. This is, probably, one of the biggest leaps of faith into the Great Unknown that you will ever experience.
I suggest that you set very modest goals for yourself. I suggest that you set the goal of meeting your brother face to face, then let things move along at their own pace. Go see the guy, take along your own contact information, then hand it over. Include a note to the effect that you are offering the hand of friendship, but you would understand if he doesn't think that would be a good idea.
Having made the first move, wait. Just wait. It might be a good long time before he feels up to making the second move. It might be never.
You don't want to badger the guy. You don't want to force him into an unnatural, difficult relationship. Just make the first move, then wait. It might be the "right" time for you, but the "wrong" time for him. Maybe he's involved in a romantic relationship that sucks up all his energy. Maybe he's in a life change situation and needs new family like he needs a hole in the head. You just don't know.
I can't tell you what will happen. I can only tell you that you need to be respectful of his space and his privacy. Make a move. Wait. See what happens.