I think I'm probably the most qualified to answer this, as I have dated a super hot multi-millionaire.... although that's not a story for this forum.
Basics - 1. You can't need her. 2. You have to be ok that you might not be her type, and no amount of "game" or fancy lines will change that. You have to be ok with this. 3. Her being a princess WILL come up in conversation at some point if you do get into a conversation. When it does, just say "Cool" with a half smile and a shrug, and change the topic of conversation. Don't stay and talk about her life as royalty, change the topic of the conversation. 4. If bodyguards come, she is the only person who will be able to call them off. Smile at them and make eye contact, Say "Hey" to them without expecting a response, and go straight back to talking to her as if they weren't there. Don’t try and make them like you, you’ll loose her interest. 5. Don't kid yourself, it is unlikely that you will be able to pick her up. She's a princess. But there is a chance you could, so go for it!
So for meeting a hot, young, princess, the only way I would recommend opening her is direct, and during the day. Why? It's the only thing she won't have experienced before. It will instantly trigger thoughts of the princess in the love story, like Aladdin or something. You will have your cards on the table about your interest in her, which is then less likely to spook her. Going indirect will likely cause her to be quickly thinking about what you want. And her being very rich, she will have on her mind the horror stories - e.g. abduction. So go direct and honest, you don’t want to spook her.
I'd see her out shopping or something, preferably when she is
not with any friends, go up and stop her and say:
"Hey... this is kind of crazy but.... I just saw you from over there, and I thought you were really beautiful." (pause for a micro-second, smile, hold strong eye contact, put both hands in your pockets and take half a step back. "You look like a sagitarius" She may or may not jump on this conversation topic. If she doesn't, be prepared to explain that someone taught you a little bit about zodiac signs and how they relate to personalities. She looked reserved and like a deep thinker, and that's what Sagitariuses do (This is totally made up. Anything will work here as long as your relaxed - e.g. if she says no that's wrong, shrug and say that's just what you were taught).
From here I recommend you talk about the following things in this order:
1. School / subjects she is learning
2. Her future ambitions
3. Her favourite films / food / whatever
I would then try either for an instant date (“hey, well I was just going to go get a coffee. Why don’t you come with for 10 minutes?”), or if she’s busy, go for the number. “OK, well I have to get going, I’ve got stuff to do. But I’d love to invite you out sometime” Then shut the fuck up. If she says yes, then say “Ok cool, what’s your number”.
If she says no, smile and say “that’s fine, no worries”. The tell her: “Well you’re really cool. Me and my friends are going to do XYZ on Thursday, you should come - just as friends”. This then gives you a second chance

hit on her unashamedly.
And if she’s not up for that, then you’re not her type. Walk off with your head held high. You took your shot like a man.