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SOLDIERS: Target the Medic.
Unless you get a good combo going on the Soldier, you can't outdamage the Medi Gun. As soon the Medic is gone, every damage you do can't be healed, so you can slowly chip his health away or finish him off with some good hits with the Lawnchair.
MEDICS: Stay behind your meat shield.
The Soldier is your meat shield. You can heal him, but you can't heal yourself. Stay at a distance from the fight, so splash damage does not hit you. Always stay behind your Soldier so he can block attacks for you.
SOLDIERS: Take and use the high ground.
If you defeat the enemy duo and capped the point, don't stay in the pit. Take the elevators back up. If you are in the pit, you have an enourmous disadvantage.
If you are on the upper ledge and the enemy is in the pit, you have a lot of chances. Jump off the bridge and bomb the enemy Medic, or just use the splash. The enemy Soldier should have a hard time hitting you, but don't camp.
MEDICS: STOP ♥♥♥♥ BATTLEMEDICING
Feel free to whip out your Bonesaw or your Syringe Gun when you are outside the medi gun range and deal some damage on the way back, but your primary goal is keeping that Soldier at 300 HP. Stop running into the pit like an idiot trying to take down the enemy duo with your syringes. Seriously.
Stay together.
The tournament is called "Ultimate Duo" for a reason. When you are alone, you have no chance against a duo, especially the Medic. If you have respawned and your partner is dead, too, it's often better to wait for the other guy instead of running out and trying to do something alone.
Stay alive.
Goes together with above. When your partner dies, run away. If you are a Soldier, you can just rocket jump away, with the Medic, keep shooting Syringes while backpedaling. Fell free to take some potshots, but your goal is to avoid being damaged until your partner is back.
Look at the damage, and choose your weapons wisely.
In the Advanced Options, you can enable an option to see damage done above your target. Now, a Soldier has 300 HP when fully overhealed, while the Medic has 150 HP max. Let's say you get two rockets on the Medic, and both say you did 72 damage. A little bit of math later, you know the Medic has 6HP left ( abit more actually, since he regenerates, but that doesn't matter), so just get your Shotgun and finish him off. Knowledge about how much health your enemy has left is very important.
Also, consecutive reloads are faster than firing the first rocket right after it reloaded and reloading another one. Try to cool down for a bit and watch your enemy to see how he moves, while the Launcher reloads the full clip.

THE BIG EXCEPTION: CLUTCH PLAYS AND LOW HEALTH
This is the big one. This tip is combined with the "Look at the damage" one and overrides "Stop Battlemedicing", "Stay together", "Stay alive" and others when in effect.
Let's say you know the enemy Medic and Soldier are both very low on health, when suddenly your Soldier goes down. With the knowledge that both are very weak, you can go Battlemedic them and finish them off. You usually have a big disadvantage, but it's countered with the easy kills.
Another one is a Clutch Play, when things are very close. Let's say both team's timers are currently in Overtime, and the enemy is about to cap your point, when your Medic dies. Don't run away, it's time for a clutch play, because even if you are at a disadvantage, if they cap the point it's over and you only have this chance to block it. ♥♥♥♥ disadvantages, (try to) acquire kills.