I hate to tell you this, but they make drywall corner bead in plastic for just this reason. The problem with metal corner bead in high moisture areas is the fact that a)they are made from cheap galvanized metal that is prone to rust in such conditions b)joint compound has water content that you are directly applying to the metal right out of the gate c)the fact that your contractor reapplied the same corner bead and you know it's rusting right now as we speak.
If you're just wanting to fix the issue at hand, yes you can recoat the exposed metal with primer, but there typically are no rust inhibitors in most primers and thus will have the same problem eventually. Priming simply preps for paint. Spot priming is only ever good as a start to newly repaired ares, and should always have the entire area redone. When you only sopt prime, the repaired area will tend to 'flash' or have that annoying shiny spot.
Long and short of it (and I hate to say it) is to call back your contractor. First, for not finishing the area to begin with(because I'm sure he didn't do it for free?!?) and second to replace the metal corner bead with plastic. That will virtually solve your problem forever....

Thank you very much for your very helpful response.

I called the contractor and left a message. It would be very hard to ask him back to replace the metal corner bead with plastic. He said he would clean the mildew above the old sheetrock when he did the free estimate. Later his crew member did not do it. Afterwards he explained there was no mildew up there. He said he would take the old insulation above sheetrock out since it had mildew on it and put new insulation on. But on the day of the work, he did not bring new insulation layer, saying it does not make any difference.

I checked the Zinsser Bulls Eye 123 primer, it said it's RUST INHIBITIVE. Would two coats of the primer help preventing the metal bead from the possibility of getting rusted in the future?

Thank you very much for you advice on the priming. It's lucky that I haven't got it started yet. The painting contractor came yesterday, saying it's only necessary to spot prime the new mud area.

I feel those professionals are not trustworthy. They just want to do the job quickly, make money and leave.

The contractor part of me says it wouldn't be that hard to tear out the cornerbead and replace it with plastic, being I get paid for what I do, no matter what it is. Cut out the metal, stick in the plastic and mud it. But anyways... You may want to consider searching for a new contractor/handyman type, based on your recent experiences.
If you have/had mold or mildew on the insulation, you may have other issues above the ceiling line. As far as 'cleaning' mold/mildew, you can really clean mold, you can only really kill with bleach or equiv. or remove the dryall. I recently had to do the same thing in my own house. I had noticed a bit of a sag in the drywall by a wall. After doing an investigation, I had discovered a condensation problem coming from my chimney that was probably going on since the house was built. I cut the drywall in that area and discovered a mold/mildew condition that was right under the ceiling joist, which I probably would not have discovered without doing my little investigation. You do not want ANY kind of mold or mildew in your house, it can cause or trigger a number of medical conditions(asthma,allergies,etc.) especially in children or the elderly. The room I was working on is for my new baby girl we have on the way. See the concern....
To answer your question of using the Bullzeye; any primer will slow the rusting process, but will not prevent it forever unless you find another way to vent the room to totally prevent any kind of condensation build-up(which, of course, is next to impossible). Being the skylight is boxed out,like they normally are, it will have the tendency to collect in that area. Even though I can't see the repaired area, I still stand by the not just spot priming thing I had said before. You will definately will see a difference. Keep in mind, this is your house, not the contractors. Unless you're okay with 'just okay', can't see it from my house in a snowstorm theories. Not to be mean, but I think you understand my point. That's why guys like me can go into a job and charge top-dollar, because I treat every job/house like my own. My company(and personal)creedo is do it once,...do it right.
I hope I helped you in some way. I really hate seeing people being taken advantage of. If there is anything else I can help you with, please write and I'll respond ASAP.--Dave