It just raises the max. When you reactivate energy, it seems to give you a short grace period and give you back some or all of your energy between 100 and 150 depending on when it ran out, but I don't know exactly how the extra energy is determined.
Premium character is about the same with soldiers, but I've only activated it three times in the past year. The only time it accidentally ran out, reactivating it did fill my hp to the premium max. Someone who's had it run out more can tell you better than I if it's prorated.
With fort bonuses, it seems you just need premium character when the battle starts and if you activate it during a battle in progress, you don't get the extra bonus during that battle, but if it runs out during a battle, you still have it for the duration of the battle.