OP is here for only one or two orbits,so no reads.
Turn is a tough spot here and mostly I am looking for suggestions of how one could play the hand differently. It's much easier with reads and stats, but still if we are in such a situation?
Try to keep pot small? (May be even check flop, though I don't like it.)
Check turn here? What I do if he bets?

Seat 3: hamselvkar ($79.60 in chips)
Seat 5: s_malby ($613.40 in chips)
Seat 7: d poer ($448.30 in chips)
Seat 8: Hero ($846.50 in chips)
Seat 9: kojecabuto ($386.00 in chips)
ANTES/BLINDS
Hero posts blind ($2), kojecabuto posts blind ($4).

PRE-FLOP
hamselvkar calls $4, s_malby folds, d poer calls $4, Hero bets $22, kojecabuto folds, hamselvkar folds, d poer calls $20.

FLOP
Hero bets $44, d poer calls $44.

TURN
Hero bets $111, d poer bets $380.30 and is all-in

I fold here on the turn to his push with no reads. As far as playing it differently, I think checking the flop is criminal of course, but I like the way you played it best. You could have CRAI on the turn for another way to play it, or to keep the pot small, check/called a turn bet, although I think that is a pretty ugly idea as well. It seems like a pretty straightforward hand against an opponent you have no read against, and you played it fine.

You're not getting much action unless you're beat. There's the light chance that someone who loves JT/T9/89 is committed that you have AK. But 55/77/88 is most likely. TT/99 is raising that flop and I think 66 is, too. But I like to keep the size of the pot in my hands and make a similar bet to yours, but a little less and fold to a push.

Getting 1:3 in pot odds is tough to fold. We need to be sure that he has a set every time to fold. I even think that we are almost priced in if he has 2 pairs here.

Anyway I am more interested in avoiding this spot than of a call/fold question. Though I think that folding is better here.

What do you think is beating you here, Zmej? A set, two pair, a straight draw? What SD calls you preflop? 64, 69? Did he call preflop and on the flop with J9 or JT? I don't think so.
87 or 76 might be a possibility but this feels like a set and he's making you pay for the FD in case you are on a suited draw here. I think you might see a bluff here because he might feel much of your range either missed or is an overpair with few outs to improve. How is your image since he's been at the table?

Maybe you bet $35 on the flop, rep a big A and see whether he plays it faster or not. You could also check the turn if you don't plan on calling a push.

Calling his push as played means you need to win roughly 30% to break even. If you planned to call a push on the turn you could have bet more there so you get a better rate on the call of a push. If you bet the pot rather than $111 you change the numbers but not significantly. Your break even win rate becomes 27%, probably not significant enough to turn a fold into a call here if you are inclined to fold.

I think you will see sets and 2 pair here more than you will see hands you beat. I wonder how often he will push here with less because he then is putting you on a bluff with your turn bet if he pushes with hands that don't beat overpairs. If you believe you are winning more than 30% of the showdowns here you got the right price to call.

Outside of overbetting the flop (I don't like it) and the suggestions above, I don't see a way to significantly change the way you played this. I do think you have to call here at least some of the time just to keep from getting run over.

Pros of checking- Worse hands will bet.

Cons- They'll keep betting like they have the best hand whether they can beat Kings or not and when you're beat you spend a lot more than $111.

Like I said, I'd bet less like $99. you're getting ~2:1+ (?) 620:280.

55/77/88- 9 combos we're damn near drawing dead
78- 9 combos we're 5:1 dogs

I just think his limp/call+flop call shuts down his range where all I really see us beating is T9 and JT for 24 combos, but I think he's raising the flop in position with these or flat calling the turn. We're 3:1 favs against T9 and 5:1 against JT.

When we're beat right now we lose 280 15/18 times
When we have the best hand we win 620 19/24 times, but I think he flat calls the $111 with these hands less than 33% of the time so it's more like 6/8 with that weight for:

9 wins= 6200-620= +5580
20 losses= -5600

So he has to push instead of flat-call JT/T9 more than 33% of the time and I just don't see that with no reads.

EDIT: My math could be way off. I've just ended a 10 hour session and I've been drinking.