I think you're giving him way too much credit for a strong range, when he's gonna have worse than two pair the vast majority of the time. First off, he has an odd stack size, which indicates a weak player right off the bat. Next, he checked the flop meaning he most likely did not connect with it. You didn't give any specifications, but I think most micro stakes players would c-bet a flop like this if they were holding KQ. 

That makes his range full of Kx's. There's also the odd J10 that you could be facing sometimes, but there are so many more one pair hands in his range (even lower two pairs, K4, K2, etc) that I would never play so passively. 

If I were to play this hand, I would have 3bet preflop. You have position against a weak player, and by three betting with what is most likely to be the best hand, you can get him pot committed and stack any one pair hands. 

If you're not going to 3bet preflop, you have to lead the flop. By leading, you protect your hand against the odd straight draw, and charge any draw/one pair hands looking to improve. 

After that, it all depends on what you know of your opponent. I don't think a flat is bad on the turn, since it can induce a bluff on the river. On the river, I would think about raising, as he is pot committed and will call with any Kx, slowplayed Ax, and even a pair of queens. I don't think that the J10 straight is a big enough part of his range to limit ourselves to only calling the river. He has more bluffs than value hands, and we beat most of his value hands as well.

Yeah, you are giving a random guy who is most likely just clicking buttons way too much credit. 

3betting pre would be fine too, but if you are uncomfortable stacking off with it for 40bb then its not like flatting isn't profitable. 

As played he can probably play KQ/A4/A2 (and K4s/K2s if he raised them preflop) like this for value and definitely still have some bluffs. If he didn't bet full pot on the river I would always raise, but since he did he is more polarized meaning he probably has very strong hand or nothing so you can just call.

Either way you should definitely never fold here.

BvB I definitely 3bet AQs pre as my default line, and call it off if villain wants to play for stacks given the eff stack size here. The only reason I would consider changing this is if villain had folded to a ton of 3bets and it was +EV to call IP and play post-flop

As played I think I'd prefer to bet the flop and expect to get value from Ax hands, FD and some combo's. I'd be surprised if I could find a fold here ever. 100bb deep that may change 

Just my thoughts :-)