I'd say just be direct. There's no need for something elaborate or cute. Just a simple "Hey, would you like to maybe [get some dinner/see a movie/go out for drinks] this [Friday/Saturday/Whatever]?" You could be more specific if there's something you know he'd be interested in, like if you were both talking about a movie you're both interested in, or if he expressed interest in some restaurant you mentioned. Just be lowkey, casual. That is, don't make it some big dramatic declaration. That puts him in an awkward position. It's not a marriage proposal. It's just an invitation to spend some time together. Basically, pretend you're inviting your best friend out for some activity. That same tone (but choose an activity that is very clearly "datey" to avoid the whole "is this a date or not???" situation).

On the possibility of rejection... don't take "I'm busy that day" as a flat "NO, NEVER." Some people just are legitimately busy, but in the moment it can feel like "HE'S LYING JUST TO SPARE MY FEELINGS, I'M GOING TO GO CRAWL UNDER A ROCK!!" Try to adequately gauge his interest, and pay attention if he says "maybe another time." You could also try suggesting another time yourself (try to make it concrete, don't say "some time"), but multiple "I'm busy"s could be bad, idk. Depends on how busy you think he actually is, if he tries to suggest other dates, etc.

And if he does just say he's not interested... it's not the end of the world! It's really better to ask and get turned out than to just spend the next few months wondering about what could have been. So no matter what he says and how things go, you're best off asking. It's a good experience, it's good to know, and you really don't have anything to lose. It's a bit embarrassing, even disappointing, but keep your composure. Just smile and say something like "ah, well, didn't hurt to ask" and move on. Keep chatting with him if you feel comfortable with it, and don't avoid him after. It's totally fine and doesn't mean there's anything wrong with you.

I say all that not to imply that I think you will be rejected. I have no clue. But I know for me, as an anxious, shy person, it helps to be prepared for the absolute worst outcome. Once I can convince myself the worst outcome is survivable, and not even that bad, it eliminates a lot of the fear going in.

And I will say that I did ask a guy out once, back in my high school days. He turned me down politely. We continued to be friends, never spoke of it again, and he didn't act weird around me. Basically, I survived. With a slightly bruised ego, but overall better for the experience.