In Blind Pick, the first player to choose a champion gets it. Your only options are to talk to the player and ask if you can play that champion instead, or to get a faster computer that gets you to Champion Select faster. :)
You can also have better luck if you focus on a less popular champion that not many other folks play.
As far as your masteries and runes go, you can typically build multiple general-purpose rune and mastery pages. For instance, some possible general rune pages:
- Ability Power, Magic Penetration, Health - good for most AP champions
- Attack Damage, Armor, Magic Resist, Health - good for most AD champions
Some general mastery pages:
- 21/9/0 with all the Attack Damage stuff in the Offense tree (good for most melee & ranged AD champs)
- 21/9/0 with all the Ability Power stuff in the Offense tree (good for most AP champs)
- 9/21/0 (good for most champions you might want to be "tanky")
- 0/9/21 (good for most support champions)
You can then switch to whatever general page works best for the champion you got, even if you didn't get the one you hoped for.
These won't be as perfect as mastery/rune pages that are tuned exactly for a champion, but if you're working on learning a champion they'll be pretty good. Until you get really experienced with a champion it's unlikely that the tiny edge you're getting from a precisely tuned mastery/rune page will outweigh your personal skill level anyway.