This job is going to fall into the "whatever the minimum charge in your area to get a skilled contractor to show up" is. Around here (Cambridge, MA), that's a $200 job for a non-handyman (skilled plasterer), maybe $100 for a handyman.
By comparison, I had a whole room done in blueboard with skimcoat plaster for $800 cash, probably 20x the area of your job at only 4x the cost.
Small jobs the guys will spend more time and energy scheduling and coming to the job than doing the job and you're paying for that (or they're not coming).
If you're replacing the whole ceiling in an area, that's a pretty forgiving job in terms of not having to critically match surfaces and get the joint compound perfect. Youtube will have many videos on how to do drywall. Fundamentally, it's cut, screw, mud/tape, mud, and sand. The skill comes mostly in the mud steps. If you're bad at it, expect a lot of sanding. IMO, the specialty "knives" (mud tools) are worth it. For your job, get an inside corner knife (90* blade) and a ~4" blade. If you have to join to another flat piece in the same plane, also get a wider knife (8+ inches). Trying to muddle through with only a 4" straight blade will cost you a ton of time in sanding and you'll probably still not get a great outcome.