To be honest, I think preflop is debatable with A9s against a broken stack. If you are somehow mixing it as a 3b/fold, its fine.

Post flop seems fine. The A9s with the BDFD are going to be the best bets here on the flop. I would mostly check A9cc and mostly bet the other 3 combos of A9s.

Seems fine.

Pre is fine, but at low stakes where rake is high and you potentially are not playing well you could maybe fold I guess. Something like A8s should be close to 0ev theoretically, so if you're making a lot of mistakes and rake is high, then A9s might turn negative, but defaulting to 3betting with this hand is likely going to be a good idea. Sizing for your 3b is on small side, but because he has a shorter stack it makes the sizing actually good.

It should be noted that when playing against weaker opposition their fold to 3-bet when they have position is generally speaking going to be low and they will also not have many/any 4-bet bluffs. Our opponent having a lack of 4-bet bluffs allows us to 3-bet slightly wider. Conversely, the majority of our EV when 3-betting OOP for our weaker hands (i.e. weaker suited hands, weak aces, weak pairs, weak suited connector hands) comes from our opponent folding preflop. I wouldn't really put A9s in this bucket (though it is close) under the assumption our opponent is likely calling all of his weaker aces. That being said the weaker hands in our preflop 3-betting range would include weaker aces, hands like QTs, 66/77, weak suited connectors. Those hands would lose EV assuming our opponent is overcalling preflop and also played perfectly postflop. Under those assumptions our weaker hands would go from very slightly +EV to likely negative.

Because we're in a 3bp with a linear range and on a very dry and static board most of our strong hands will continue to stay strong. There are not any flush draws, 2's shouldn't be in our opponents range, and there are not many relevant possible straights on future streets. This leads to a situation in which we are relatively polarized from the flop and the likely preferred and optimal sizing will be close to a geometric one. Geometric means that it will be a sizing that allows us to bet the same % of the pot on all three streets (or two streets) to get all in by the river. This maximizes the EV of our range in situations where these variables are true because it forces our opponent to defend the maximum number of combinations while also retaining our ability to get it all in by the river. Because the stack depths are short in this hand you have the option of a geometric sizing to get it in on three streets or over two and both are probably good strategies for most hands in your range.

Betting the flop is good and you'd probably want to bet all/most of your range vs. a player that is likely wide from preflop.

As played on turn, you actually have decent showdown with your hand vs. some weaker hands that would have peeled the flop. So this is sort of a classic spot where you won't get better hands to fold and you won't really get worse hands to call. When that is the case typically checking is usually a good idea, so checking turn is correct. Now, facing the bet I think you also have a pretty easy fold. A lot of weaker players (though certainly not all) generally opt in to a more passive play style where if he did call the flop with some hands like K high they don't always bet when checked to on the turn, so when you face the bet on the turn there really isn't much to do other than to fold.

All in all, well played!

if he is not topping up then he is not competent, so he is either raising very tight or very loose here

with no hud stats you ought to think about if 3betting is good vs those two tendencies and if your 3bet is +ev in both cases

vs very tight your 3bet is going to be close to 0 ev at best, vs very loose your 3bet is ok but you are better of merging and 3betting AJo/KQo

calling vs very tight is -ev, calling vs very loose is probably slightly ok but 3betting is better

if you put that analysis in a 2x2 pro vs con chart then its easy to see this calling is the worst option so 3bet or folding are your choices with folding being just slightly better

but you elected to 3bet which is reasonable and cbetting this flop is going to be +ev in a vacuum and check folding turn is fine, but with no reads we cant really learn anything from our line nor how to adjust since we dont have a baseline for comparison

your on the right track on the turn with your reasoning - you have no idea wtf is going so you check fold

but it seems like you had no idea what his range was preflop as well with no hud or reads, so use logical continuity and fold there as well until you have the information to deviate