I really want to fold this preflop, but you've already stated that.
However, I also badly want to fold it on the flop.
On the turn, going for a checkraise is fine, but I would have liked it to be a bigger raise. Half the pot gives him 3-1 if he's drawing, and he's in position, granting him good implied odds
The river? Ew. I hate that he raises us. We have no idea what he's holding, and even though we can get paranoid (set, flopped straight) it's at least exceedingly unlikely that he's holding TT or QQ, so two of the five sets he could have are not going to happen. There's too good of a chance that he's holding a weaker two pair than us (Q-4, etc.) or simply an open-ended draw with a Q-kicker that figures he has us beat now that he spiked top pair on the river, not to call here.
I don't really want to add up all the bets to find out the pot size, but it looks like somewhere in the vicinity of 11-3 or so, and the chance of him having a weaker hand is often better than that. Go for a call. I'm not sure I'd like a raise here, though, given that we have serious difficulties putting him on a hand.