I think this spot benefits a lot from planning.
The flop hits a standard villain OOP flatting range [15% - 50%] a lot.
http://www.propokertools.com/simulations/show?b=8c+Qc+Jc&g=he&h1=Ac4s&h2=15%25+-+40%25&s=generic
Villain often has a pair, straight draw or worse FD here.
This is offset by the fact the board is monotone which devalues the straight draws and will likely cause villain to play pretty fit or fold vs cbets, improving your fold equity.
When you decide whether you will cbet, you should have already decided what you will do if villain [raises, jams, flats].
If villain jams on you, are you calling off?
What if he c/r NAI?
How will you play the turn and river if he just flats?
 
Against top pair (crude benchmark for getting it in in hypers - somewhat less relevant on this VERY wet board) you have FD = 9 outs and an overcard = 3 outs
12 outs ~ 45% equity
Mandatory call off if he jams at this stack depth once you account for the money already in the pot.
Also an [A] might not be a clean out vs some pair + gutshot hands which could reasonably get it in here but it will probably be offset by the small amount of times villain stacks off with a worse bare FD.
What about checking back?
Most of your equity vs villain's range is from your outs, so taking a free card benefits you more than him.
AND your opponent is rarely going to fold a worse flush later (or crushed 2 pair or straight) anyway so you won't lose value by not building the pot now.
By checking back you flop keep the pot small against a range you're at best flipping against, keep some bluffs in his range and leave enough money behind to give yourself implied odds to call a turn lead (dependent on sizing).
You will have position on the river and should be able to make a profitable [jam, check down, fold] depending on how the board runs out and your opponent's action.
For these reasons I prefer checking back the flop to cbetting.
 
But it's pretty close and probably not a spot which will make a lot of difference to your expectation over time.
 
When villain leads a blank turn 0.5P then checks on an overcard river his range is probably mostly [Qx, missed FDs] but more heavily weighted to Qx because he might bluff some of his missed FDs. Your pot size bet needs to get folds 50% of the time to be profitable.
 
He was pretty stubborn calling down with such weak relative hand strength, but you also have a lot of missed draws in your range after flatting the turn lead and are somewhat polarized in this spot (if he's even thinking about your range).
 
Again it's really marginal but I think checking back the flop helps improve your overall expectation of the hand because you will win more the times you are ahead and lose less the times you're behind. I think the marginal expectation gained from your increased implied value (leaving more money behind) on the turn more than offsets any lost fold equity from not cbetting.