[Melchiades]

Bet the turn. From your description of him, he knows you could make that bet on the flop with something or nothing. He likely therefore think your check behind on the turn means you missed and have nothing. His bet on the river could therefore either mean he caught a piece of the flop or he now thinks his 33-99 is good. Or do you think that he could be betting with AQ,AK,KQ thinking you missed and cannot call ariverbet?

I most likely fold on the river. It comes down to how likely it is for him to bet out with A high or something here if he thinks you missed, and how likely it is he has a J or a pocketpair. A raise on the river will only be called by hand s that has you beat. It will look like a bluff after the turncheck, but he doesn't sound like the kind of guy to call with AK just because he thinks you are bluffing.