A lot of grad programs require you to take an entry exam, and also write-up an essay to send to the grad advisor.
Self-study won't necessarily help you pass the entrance exam. Depending on grad program, you often get stuck with a generic GRE exam that just ensures you know basic stuff you should have learned.. like reading, writing, math, etc. Some programs have had issues with foreign student language, so they may ask a foreign student to also pass a language test. (The tests must not be very hard, because I had quite a few exchange students in my grad program that could barely speak English.)
Where your self-study will shine is if you keep a portfolio of projects / work you do on your own, and you can reference that in your entrance essay.
Grad advisors are like hiring managers at jobs. They see tons of run-of-the-mill essays that just try to BS. But, if you can write an essay that specifically references things you worked on, are working on, etc, the grad advisor might be more impressed.
EG: I applied to info sys grad program at my college. I had years of experience in the working world as an analyst. So, in my essay, I explained how I had focused my career on doing info sys work, how the undergrad had helped me, but also how, because of my experience, I knew the undergrad was not enough to take my career to the next level. So, I really wanted to shoot for the grad program to do that, which had more focused studies. I pointed to several research projects I did on my own to help show that I was serious about what I did... I quit my job and went to college full-time to get my degree(s), too. Grad advisor was impressed with my work and essay, because it stood out.
For a Physics program, professors and advisors would probably be more interested in someone that's so into Physics they like to mess around with it in their spare time. So, self-study could help.
Self-study would also make it easier transitioning into a grad program, because a lot of times the classes start with a refresher. You'll be able to reinforce knowledge you already have, and can possibly have a leg-up on others b/c you already know some of what's going on.
If you're serious about pursuing physics, then do a project portfolio. It can't hurt, and it can only help. A lot of careers are wanting folks to have portfolios of work now.. not just art folks, but programmers, analysts, etc. It's just the direction we're all going in.
There's mixed signals about this, though... Hiring Managers say they're interested in seeing someone that has personal projects; shows the person is dedicated to what they do. But, recruiters and hiring managers also say they barely glance over a resume, let alone side projects someone has done. So, it can seem pointless.
So, my opinion about personal projects is to work on ones that you are interested in and have fun on. When publishing a project, explain how it can apply to real-world application (applied science), because there's a lot of folks that have trouble taking high-level theory and translating it into application. We need people to "translate".
EG: I'm trying to implement a physically-based rendering lighting model in a video game. My head spins reading a lot of high-level white papers. But, I give kudos to folks that have taken the time to interpret that into real-world application in tutorials. If I was a hiring manager or grad adsvisor, I'd look at that and go "this person added value by translating academic speak into something more practical folks can understand."
I could go on and on about this...
But, side projects and self-study will help you get into a program if you can leverage it to show a grad advisor that you're serious about the field.
Some colleges are bypassing GRE's, and just letting folks into grad school now... I think it's for certain things grad programs that are hurting for money... like MBA's and junk. Colleges are in business to make money, so as they saw less folks pursuing masters (b/c the cost is outrageous, and the benefits are not as good as they used to be), they try to entice (sucker) more folks to sign up for grad programs to since $40k of their life away.
You need to be really sure the grad program you want to shoot for will actually help you.
I got a masters in info sys, and it's been pretty useless. In the analytics industry, a degree is just a check-box. All most folks care about is that you have a bachellors. A masters doesn't really get you anything else. What folks are really interested in is if you have 5 yrs exp in specific software suites, ERP's, etc. OR, that you have 5 yrs exp as a data scientist.
So, if you love physics, make sure the masters in physics would actually help you before pursuing it. Some folks (like myself) jumped for a masters thinking it would put them ahead, and all it does it leave you in a dead zone.. b/c folks think you're over-qualified for entry work, and under-experienced for senior work.
My masters in IS would help me if I had management experience, b/c it's what CIO's, IT managers, etc need. But, I've never had management experience, so nobody's going to hire me out of the blue and take a chance on me as a manager.
You have to make sure your degree is going to be worth the time, effort and expense.