You can tell if the next SP you'll face, at any level is R or L by going to the Lineup page, setting the level and pressing Go. Then click on Lineup again. If will automatically go to R or L, depending on the next SP that team will be facing. I think if the next day is an off day it always show L, but I'm not 100% sure.
If you really want to know the exact SP you'll be facing next, you could go to the other teams Schedule page, see who the last SP was, and then go to any screen that shows their roster and figure who's starting next. Seems like a lot of effort for little payback to me, but if you're into it, it can be done.
It would be wrong to do what you're asking with starting C. Just like in MLB, both teams submit their starting lineup at the same time. So there's no way for a manager to be 100% sure who the other team is starting at C and adjusting his lineup accordingly. Why shouldn't the other team get to see you're starting a lot of fast players and then get to change their C? That's the same advantage you're asking for.