Party NL 50 full ring

I'm in BB with 88. Folds around to CO who makes it $2.25 (blind steal?) Fold. Fold. I call.

Flop comes JJ4. I check (no idea why. I think I was going for a C/R if he bet small enough.)

He bets $4 into the ~$4.50 pot. I think for a while (stacks each were about $55 to start hand), and fold.

My other option was to call the $4, then check-raise all in any turn card lower than a "7". Is that a better option? I didn't put him on an overpair. Just overcards. Don't put him on a jack...and even if he did have an overpair, I may push him off of it if he thinks I have a J.

Which would have been better, assuming you called his PF raise (and checked the flop)? I know I should have led out on the flop but for this example assume you checked to him.

Was I right to fold or would the second option have been better? What should I have done if a 9 or higher fell on the turn? Still push?

Thanks in advance!