What solder is used for soldering magsafe 2 charger outer wire?

I don't recommend opening the charger to do any service. While I know people want to save them the risks to your self and your system I think are just to great here:

• Personal electrical shock if you don't reseal the unit
• Possible fire risk
• Damage to computers logic board if mis-wired or if the wire short-out

As to the cable you got. These are from the crappy clone'ed chargers Apple prevented from being sold. Apple forced the company to cut off the cords of the charger units they had made to prevent their use. What this sneaky company then did is sell the cords and the chargers as parts!

I had gotten one of these chargers in with one of our field folks which killed his system as the sense wire shorted out with one of the power lines. Because of this we branded our chargers so it was clear which was a known good unit as the clones are so good you can't tell unless you have a real unit side by side. Compared: Fake vs Genuine Apple Magsafe Chargers

Getting back to your solder issue... You'll need to use a crimp style joint as the wires used in this cables construction uses a filament with a foil wrapped around (not the same as Apples).

Frankly, I would just get a real Apple unit and call it a day. What you save today will cost you big time later with a burnt logic board!