Shooting spitballs at the Justices is still allowed though, right? 
No, seriously, it is a great opportunity for you.
You can’t take anything into the chamber, particularly electronic devices of any kind, including phones. I don’t know what they do for general gallery spectators, but press is required to leave everything except pen and paper in the press room. They might have a visitors anteroom where they’ll make you leave your phone.
Make history.
Get naked in court. 
Bummer. Couldn’t you just picture the Supremes’ faces when, during some intense verbal sparring, your cell phone sounded off with *“Ring a dong ding ding-a doh…” *
[sub]That’s my cell ringtone. I downloaded it from the Geico website.[/sub]
Well, you’re no fun.
Do you know the state? Smith v. Louisiana appears to be an Innocence Project case, and cert was granted at the end of June. If it’s in California, it could be Martel v. Clair, which also got cert in June - I think that one is Innocence Project, but I’m not sure.
I am very jealous of your opportunity.
Something like this. It’s basically a locker room for the general public. I went to see arguments a couple of years ago. The regular visitors’ viewing area is waaaaaaaaaaaaaaay in the back of the room, though, so here’s hoping you get better seats than that!
Good sleuthing. My wife’s cousin is one of the attorneys in Martel v. Clair (David W. Fermino, based in San Francisco)
Wow, how exciting! I know the meat of the cases are in the documents submitted, but my favorite part of the process is the oral arguments. I love reading the transcripts.
When I lived in DC I headed to the court several times. If you’re interested in seeing a high-profile case (the pledge case, the early Gitmo cases, etc.) get there in the late evening, bring a sleeping bag/mat, and talk conlaw with all the other folks sleeping out for a seat. Not a good seat, just entrance into the building. If you go on a day with non-glamourous cases (which are still interesting), you don’t necessarily have to sleep over. There are large lockers provided after entry to store sleeping bags and other bits and pieces.
When I lived on Cap. Hill I used to head round every once in a while when I couldn’t stay or couldn’t sit the next day for a case and bring a pizza or other creature comforts to the random strangers camping out. Not quite like a Dead show, but some things in common.
YMMV, and I haven’t been there since '04–security may have changed.
(Ticketholders got through a different door, btw.)