So I'm a junior HS student very interested in mathematics and I think I have pretty good chances to get admitted into Stanford online HS and take a university level real analysis course that's worth 5 credits. It also says that a course on linear algebra and multi variable calculus is strongly recommended, but I have limited knowledge in those fields. What should I expect? What work should I do in order not to be overwhelmed and fail the course? Should I just switch to Elementary Number Theory?
Thanks in advance and sorry if that was too long :)
Do their recommended courses first.
It is common to have Linear Algebra and Multivariate Calculus as prequisites to Real Analysis. In fact Linear Algebra and Multivariate Calculus are usually the first Pure Maths courses any university student takes.
So the two courses are taught with that uninitiated audience in mind, working not just as an introduction to the subject, but also an introduction to the differences between university level and high-school level maths. In particular university maths has a much higher degree of rigour and (if you're lucky) will focus more on understanding than on being able to tackle a prescribed list of problems.
Since this will be your first university level course, I suggest you follow the standard teaching order. It's true that Real Analysis won't require much knowledge from Linear Algebra or Multivariate Calculus. But you will find the level of rigour of Real Analysis very hard to swallow without a more gentle introduction the other two courses should offer.
I would be cautious about starting with Number Theory. Number theory deals with ideas you probably haven't seen in high school before, whereas the basic objects of Linear Algebra -- vectors and matrices -- and calculus -- derivatives and integrals -- should be more familiar to you. Also my Number Theory course was taught in second year and was more difficult because of that, so I wouldn't guess how Stanford's one fits into the framework.
My main advice to you is to not hurry the process.