Supreme Court Visitor Services Page talks about how to enter the building and says that "no photography or video recording is permitted inside the Courtroom" but doesn't really explain how to explain how to enter the courtroom when it's in session. Can you just walk in?
All oral arguments are open to the public, but seating is limited and on a first-come, first-seated basis. Before a session begins, two lines form on the plaza in front of the building. One is for those who wish to attend an entire argument, and the other, a three-minute line, is for those who wish to observe the Court in session only briefly. Please do not hold a space in either line for others who have not yet arrived.
Seating for the first argument begins at 9:30 a.m. and seating for the three-minute line begins at 10 a.m. The locations for these lines are marked with signs and there is a police officer on duty to answer your questions.
Visitors should be aware that cases may attract large crowds, with lines forming before the building opens. Obviously there are unavoidable delays associated with processing and seating large numbers of visitors, and your cooperation and patience are appreciated. Court police officers will make every effort to inform you as soon as possible whether you can expect to secure a seat in the Courtroom.
Groups with reserved seats should line up to the right of the three-minute line.
You will go through a security checkpoint as you enter the building and again as you enter the Courtroom. Weapons or other dangerous or illegal items are not allowed on the grounds or in the building. Please refrain from taking the following items into the Courtroom when Court is in session: cameras, radios, pagers, tape players, cell phones, tape recorders, other electronic equipment, hats, overcoats, magazines and books, briefcases and luggage. Sunglasses, identification tags (other than military), display buttons and inappropriate clothing may not be worn. A checkroom is available on the first floor to check coats and other personal belongings. Coin operated (quarters only) lockers for cameras and other valuables are available. The checkroom closes 30 minutes after Court adjourns.
We do not recommend taking infants or small children into the courtroom.
When the Court adjourns for lunch all persons must leave the Courtroom and the Great Hall. Persons attending the afternoon session must line up again on the Front Plaza to gain admission.
So join the line and wait to get in how long is the line and when you should realistically arrive to get in is a different question.
I went on a day with a fairly boring case and showed up around 7:45am. I waited in line for many hours and missed the entire first argument. I made it in for the last 30 minutes of the second argument (around 11:30am). It was a great experience but I was really close to not making it.
At first there is only one line that you can get in. I would recommend getting there as early as possible but not later than, say, 7:15am. Once the time gets close to around 9:30am there will be an announcement and the formation of two lines. The line you are already in will be for a permanent seat in the hearings. The new line to your right will be for quick 3 minute visits into the hearing. Hopefully, most of the people in front of you will move over to the new line and you'll be more likely to get in.
After that a set number of people from the permanent line will be let in. There was surprisingly little turnover after the first argument ended. The process of getting is also very slow, with multiple metal detectors, instructions, locker stop, etc. If it's getting close to the end of the first argument and you're not within, say, the next 30 people in line I would jump to the temporary line and at least get a few minutes in the court. Keep in mind, you can jump into the temporary line at any point. Also, you can get right back in the temporary line when you're done with your 3-5 minutes. It also seemed like some of the temporary line people stayed in significantly longer than 5 minutes, so this might be a good option if your desperate.