You don't need to get the license rush-shipped. Print off the "permission to race" form and you're good for 30 days.
As for the expiration issue, you'll need to check the USA Cycling website for that answer. I'm sure it's on there.
You'd be purchasing a "Sport" license, correct? That'll get you into "C" and "B" cross races.
Good riding at Rutgers today, btw.