You can tell which dragon is going to spawn next, AFTER THE PREVIOUS ONE IS DEAD.
During the dragon's respawn time, on a wall of the dragon pit there would be a rune that you can easily see. The rune is the signature of the type of the NEXT dragon. They are usually pretty self-explanatory in design so when you look at it you should have an idea which it is that's coming up.
Another method is to press the "Tab" key and look at the dragon respawn counter. They will also show the same symbols under the counter that tells you which dragon is coming up next; and if the dragon is live it just shows the symbol of the dragon. And if you 'ping' it and paste the timer into the chatlog, it will tell you exactly which dragon it is in text.
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However, that's the extent of how much you can tell.
You can not foresee for instance, all rotation of the dragons the match is going to have before heading in.
I've always advocated that it would be a neat feature to have both in general play and professional play that actually benefit both experiences. But it doesn't feel like Riot is planning to do that any time soon and insist on keeping the Dragon aspect of the SR RNG based and shrouded in uncertainty for some reason.