Boxing:
Bring extra cards, because you'll be shooting more as it's 100% stand-up action, and you'll be relying on your burst rate a little more to capture the action.
Some of the better shots that capture the "spray" from the punch is right after the bell, as the corners has poured water over the fighters heads.
Used properly, the ropes can help shield your lens from the lights.
Per usual, keep that aperture wide open and that iso high, 1600 sounds about right, and the shutter should be no "slower" than 1/500 in most instances, light providing.
I've shot boxing a few times, and it's good action, but it really isn't my proverbial cup of tea. That, and it didn't pay. I guess you could say I'm a sucker for paying my bills.
MMA:
My camera has a 1.6x crop factor, and as such my prefered lens is the 35mm f/2, which I usually shoot with the aperture value at 2.5 to prevent DOF issues. Very sharp, fast lens, and a fraction of the price an "L" zoom, and plenty adequate considering I'm shooting against the cage as opposed to the "big" productions where one would have to shoot from behind a catwalk.