I have ridiculously bad cramps - they're not quite as crippling now as they used to be after my doctor put me on the Pill (I used to spend three or four days of every month practically sobbing in pain and bedridden and useless; now it's one day where I'm sluggish and in agony).
I remember being told "exercise will help!" But this is a very individual thing and it depends on what you do, too. I went swimming once and just about drowned from the agony. But I've skated on my period before and have found it actually helps. I always thought the cold would make it worse, but for some reason, the movement actually makes them go away.
If you get bad cramps, you're going to need something a little stronger than Advil or Tylenol. I used to use Naprogesic, but now I use Ponstan, which is much stronger. Beware that both of these must be taken with food or you'll wish you just had period cramps! If you've got a comp, I'd double check with the drugs policy and make sure they're okay to use.
I use a pad more often than a tampon. I use the "invisible" type pad with wings - it's literally invisible under my practice dress so I imagine it'd be invisible under a competition dress. If you're nervous about using a tampon (although with the average age of skaters on here I can imagine you've already used them) I use the "mini" or "slim" type.
(It's amazing how few people seem to know about "invisible" pads though.)
Ultimately though, if your body really doesn't want to get moving, don't force the issue.