I think you're over thinking this. You were school mates, you know each other, you've had some good conversations. Your were not really friends, but you've been friendly. There's nothing weird about sending a facebook message to casually meet up. She knows you, it wouldn't be like receiving a message from someone she has barely talked to. You've never texted before because you met at school regularly. There's nothing weird about sending a short, friendly facebook message now. It's quite common actually.
I agree that asking her out romantically out of the gate might be a bit much. So invite her to some friendly activity. If there is something that you could do together that she would enjoy, ask her about joining. For example, you know she loves a certain band and there's a concert in town, ask if she's going because you're looking for someone to go with. Or the movies. Or you can really just ask to meet up for coffee or drinks (I wouldn't ask for lunch or dinner, this gives it more a date vibe).
You also have the added "benefit" that you two will be working at the same company, so if you're too afraid to ask her to meet up during the summer, you could ask her about getting drinks after a work day to see how it's going. All of those options are fine.
If she says no, it's not because you've messed up, or are out of line, or anything like that. If she refuses to casually meet up with you, this means she's not interested in even having a friendship with you, and would have said no in person too (or say a "Yeah sure, text me and I'll see when I'm free" and never be free). If she says yes, great ! This will give you the opportunity to ask her out on a romantic date in person.
To give you an idea how such a message could look like :
"Hello X, how are you ? I saw there's going to be a vintage flea market nearby and I remembered you like vintage stuff. Do you want to come with ?"
"Hey X, how are you ? I hope it's not weird, but since we'll be working for the same company soon I thought we could meet up and catch up ? Having a friendly face there will be nice :-)"
"Hello X ! I remember you're interning at company Y now. I hope it's going well. I'm working here, so would you like to catch up at lunch or after work some time ?"
I've never been in your exact situation, but like I said, texting people after knowing them casually for a while is quite common. The last time, it was someone I was barely friendly with (more friend of a friend that I saw at parties), I had her number because she was part of a whatsapp group I'm also in. There was a band I wanted to see at a concert, none of my friends were interested, I remembered she liked them so I send her a whatsapp message. We went together and are friends now.