I am also a little rusty and got some advice from a student in the PhD program I'm attending. For analysis, I've used this YouTube series and found it immensely helpful while working through Rudin Analysis: https://www.youtube.com/watch?v=sqEyWLG ... 96F72137EC
I also found the linear algebra course 18.06 SC in the MIT OpenCourseWare very helpful in guiding me through some refreshing of linear algebra topics.
I would not memorize theorems, instead I'd work on your proof writing (formal) and really try to understand at least the basics so you can hit the ground running in the fall. I am retaking Analysis in my program this fall (I took it in my undergrad) just so I can refresh my proof writing and my deeper understanding of math and be well prepared for dissertation writing and better research.
As long as you get the basics down before your program starts, you should be able to keep up without having to go back and relearn things from undergrad. So for the rest of the summer, focus on big topics in Analysis, proof writing skills, and important linear algebra topics like vector spaces/subspaces etc.
Good luck! Glad I'm not the only one studying this summer just to be prepared for the PhD program.