No, Bards and Sorcerers are not different from other casters in 13th Age in this regard. Just like everyone else, they have to pick their spells for the day.
The concept of "spontaneous casting" doesn't quite apply to 13th Age, as its spellcasting model is more akin to D&D 4e than 3e or Pathfinder. Spells are effectively powers that can be used at-will, once per battle or once per day (sometimes with a chance to recharge).
However, there are a couple of cases that come close. First, wizards get a utility spell they can prepare, which acts as a "container" for any utility spells. Effectively, it's a spontaneous slot for utility spells.
The closest to a spontaneous spellcaster 13th Age gets is Chaos Mage from 13 True Ways. They do not select spells, instead they have a limited amount of per-battle and daily spells they can cast before a heal-up. However, at any given time they only have access to a very small sub-set of all possible spells, determined randomly.
Another example are terrain caster druids, a very flexible class. They also get a limited number of daily spells, but the actual spell list from which they choose is determined by the type of terrain in which they find themselves.