Who should repair this water damage?
The source of the problem is that's an uninsulated duct running in unconditioned space (the garage). Every time your AC kicks in, the cold air cools the metal which then condenses the humidity in the garage. That humidity runs down the duct as water and soaks into the drywall causing the damage you're seeing. 

The fix will involve:


removing the duct
repairing the drywall
reinstalling the duct with insulation to prevent this from happening again


Whether you should do it or hire it out would depend on:


how comfortable are you with some mechanical skills and learning new things?


you indicate that you don't have experience. These aren't difficult things, but doing a nice job on the drywall will take time and patience.
you'll need good spacial relations to be able to visualize the repair necessary, draw it out, estimate your materials (hello 3rd trip to Lowes this week today! don't ask me how I know), cut, fit, fiddle, install.

how accessible are the other ends of the duct? 


where it disappears into the walls - you'll need access to the top at least
the bottom looks like you can unscrew the vertical just above the elbow

how comfortable are you with drywall repair? 


It's a garage, it probably doesn't have to look too pretty, though someone may balk at an absolute bodge job when it comes time to sell the place

how comfortable are you with replacing bits of the duct?


you may need to extend the elbow out to get some insulation wrapped around the vertical piece. A pro probably would, if you DIY, you might get away with "adjusting" the existing piece a bit.
you'll probably want to insulate the elbow, too, or it will condense and drip
you'll need to get insulation into the section that goes into the ceiling



If you hire it out:


A GC isn't a bad option, but would probably be the most expensive


If he doesn't have the skills, he'll handle getting an HVAC guy and a drywall guy and getting them coordinated. 

A general "handyman" type repair service would probably be good, too.


be sure to ask for some references and check 'em - you don't want to spend good money on more bad work.

You could be your own "GC"


hire an HVAC guy to take the duct work apart (he can handle getting into the walls as necessary)
rip out cut out the damaged drywall back to the nearest stud where you've got dry, undamaged material. (This will save some time for the drywaller and probably some expense. He'll adjust your cuts as he sees necessary.)
hire a drywall guy to repair
call the HVAC guy back to reinstall the insulated duct




The more I look at your pictures, the more convinced I become that this is concrete/block, not drywall (hence your mention of "mason", I suppose). In that case, replace all mentions above of "drywall guy" with "mason".

Personally, I've done a lot of projects on my house, but I'm not certain I'd want to take on this level of masonry repair, especially as a "first house project". Others may indicate that it's a fairly straightforward masonry fix. If so, I might feel confident enough to take on, you may too...

Based on the last two images added, that's definitely plaster over drywall. They drywall repairs are simple enough. Doing the plaster on top is a bit more work and skill, but you've got the fact that it's a "rough" and "random" pattern working for you - it doesn't have to be perfect to blend in well enough, and if you really don't like it, you can sand the plaster back off (making a huge mess) and try again (just don't sand through the paper of the drywall).

The most important part is finding the source of the leak and getting that fixed. Until you do that, the rest is just putting a bandaid on a sinking ship. If left unresolved for too long, you'll end up damaging framing, not just the drywall. If that happens, you're in for a fairly large repair bill. Much cheaper to track it down now, even if that means hiring someone.