I would usually play the hand the same way up to the river. I might value bet turn 33% to 40% and even value bet river on good rivers such as this one. 

After checking back the turn, i think that we are getting a good enough price to call the river. We may not be good 50% of the time, but we only have to be good at least 34% of the time to make this a break-even call. I think he value bets enough 9x and can show up with total air sometimes. When thinking about how many combos of Ax vs 9x we have to consider the fact that there are two aces left and three nines. We also have to consider he is peeling out of the big blind vs a button open. This allows him to have a ton of 9x such as K9-98 and more suited combos as well. We should expect him to 3b at least AJ+ here in order to get it in preflop so that reduces the combos of Ax in his range. Take into account that he can have just total air leading the flop trying to take it down, folding river seems ridiculous.  

Had he bet smaller, I would snap it off expecting 9x more often than Ax. If he had bet bigger, I would usually fold to this line, but could find a hero call with reads.