That pompous White Fatalis! X_X

Jesus! I'm still with my fellow hunter friend and it made a huge difference, but here now we stand.... THIS far, just before the only "undefeated" boss of the game: White Fatalis! We have him scarred across the chest and even the eye. Third round but this time he refuses to run via time over, so I expect this is where we kill him; however, no matter what we do (I'm the girl with the rocket launcher (Heavy Bowgun) or bow while he uses the best anti-dragon Katana he has on hand. The problem?

Fatalis keeps jumping into the Thermosphere Zone and bombing us with bolts of doom! He repeats it every time I end up showering him with shotgun spread fire or if my partner hits Fatalis hard enough; this repeats till either I keep getting one-shot killed or time runs out!

How do we stop him from doing this?

My stats usually:
Main weapon A: Evil Dark Wind Cannon
Armor: Kirin Type-S set

Main weapon B: Exterminator Bow MKI
Armor 2: Full Healer Type-U set

You playing Monster Hunter Freedom Unite correct?  If so you could always just get some G class equipment.  Either way, I have a suggestion for your friend.  A very effective way to fight White Fatalis is to get his hands on Eternal Schism (The maxed out dragon dual swords in MHF2 which have overwhelming high dragon damage)  These dual swords give you the mobility and attack speed you need to go in and tear Fatalis' legs and abdomen up.  Then, you need to make sure that you activate the Extra Sharp Piercing ability (Fencing).  That way you can spam dual sword combos en masse on Fatalis without having to worry about it ever bouncing off.  (just make sure to resharpen if it ever drops below green for elemental damage output).

As for you, I would recommend the Akantor bow.  It's not as strong as its MHF2 version, but it still gets work done.  You may also want to consider switching to dual swords to use eternal schisms if possible.

If you are playing Monster Hunter Freedom Unite, the Fatalis will flee the Fatalis doesn't flee at 25 minutes, that means you haven't done enough damage to make him flee yet and he will flee if you reach that damage threshold before the end of the quest.  He will flee a total of three times, and you have to kill him on the FOURTH quest before he resets.  

One more suggestion.  When Fatalis enters armor mode (that thing where you can't pierce him anymore and he's in forever rage mode).  You should bring a ton of Large Barrel Bomb+.  They do a lot of damage and the explosion damage is not reduced by White Fatalis' armor mode the way your shots and slashes are.

Tutor pretty much nailed the armor mode work around. Another way to deal with it is if you have a gunlance with high level wide shellings. They work wonders against the presumed father of the brothers Fatali.

I'll be honest in the fact that I haven't played unite in forever and some of the mechanics the monsters use are completely lost to me. I may have to brush back up on it.