Giving my number to a stranger?

You don't even have to "drop" it - just hand it to him. If you are shy about it, hand it to him right before either of you leave the bus.

Never ever give out your # unless you are close friends that maybe use a burner phone.  Meet in public because maybe they are violent or a rapist.

It's kind of totally high school, but so what.  Breaking the ice with someone often requires a bit of cheesy stuff and awkward moments, and there are worse ways to declare your interest.

I dated a lot when I was young and single, but I took the easy route so I could avoid all the cheesy awkward stuff, and I still had to deal with the cheesy awkward stuff.  I played in a rock band, not for the money if you know what I mean, and it worked well.  But there were still countless awkward moments with girls (rock stars still have awkward moments, and groupies are girls too), because there is just no escape from awkward moments.  It's like the will of the gods or something.

While I would suggest more direct eye contact and starting casual conversation, I wouldn't say handing him your phone digits is a bad idea.  There are lots of worse ways you could do it.

It won't leave a good impression and it will make your bus rides awkward. You need to strike up a conversation and gauge his interest. If that goes well you can ask for his number, if not it's easy enough to disengage and move on.

I would be very very cautious about giving a stranger my phone number. It creates too much of risk of my privacy being compromised, too much risk of some negative outcome. I'd give him something more like an email address, a special email address that it wouldn't be a big deal to cancel the account. That why if the stranger turns out to be "bad news" you're not in a situation of changing your phone number.