I don't know the entirety of Brand's CDs, mana costs, and whatnot, but I find that usually dodging his Flaming Pillar and staying away from burning minions is good enough to survive until he OOMs. Dodging his fireball should be trivial until he gets Rylai's.
And to be honest, if you can dodge Brand's Pillar consistently, you can dodge virtually any nontargeted skill a caster champion can throw at you (except ults).
Alternatively, play Galio. Bulwark es OP.