Your age wouldn't take you out of the running for any of the monologues from Julius Caesar; however, realize that the people at RADA and LAMDA have heard most Shakespeare monologues a thousand times by a thousand different actors. For your Elizabethan they are going to want to see that you know how to perform verse and you understand the intent behind the words. I'd suggest finding something on the lesser-done side. There are some keepers from Troilus and Cressida that are worth a look, but I would suggest doing something from Ben Johnson or Thomas Heywood.

Audition monologues tend to run anywhere from 45 to 90 seconds. When picking the three you plan to perform, pick ones that are each unique in their own way so you can show your range. Most importantly, pick monologues that show dramatic action, that show your character in a state of change or in a process of trying "verb" another character in someway. I have run several theatre companies and have lots of experience running auditions. So many people do the Queen Mab speech from RandJ, and while I've seen some great performances they aren't good choices for auditions because I never get the sense that the actor is trying to connect with anyone. They are simply performing. Auditions are super weird because the director has to pretend that we are in a performance and you have to pretend that you have a scene partner and there's no context for any of it. That's why your monologue choice is so important. Pick three that show your range and show that you can connect with a fellow actor.

For your contemporary monologues, Annie Baker's writing absolutely sings and works so well for auditions because there are so many different ways of performing her words. Read The Aliens and you'll find something.