Do not approach the Man on Fire, neither fight him, nor waste time on covering from him. As soon as you cut back to your character, start running towards the destroyed hospital exit. Here, press the button for dropping down onto the ground. After you get outside of the building and land onto the ground, you do not need to fire at the approaching Man on Fire, because this is no way to stop him. You can, just as well, wait for another cutscene to start.
Now, you need to escape on horseback from the Man on Fire. During this sequence, do not waste your ammo for the shotgun (S1000), because you only have a limited amount of that and, apart from that, you can worsen your accuracy rating. Do that only when the Man on Fire comes discernibly closer, and when he is preparing to launch his ranged attack (this is required for the second optional objective of the mission). It will take several tens of seconds to lose the Man on Fire, which ends the prologue.