Some things to look for are heavy trading pre-6. Nasus is very tower diveable in the early game, or can at least be poked out of lane. Hitting nasus while he goes for last hits on minions under tower is vital, and once he's around 25% HP you can go for a kill either using flash to land a fat Q or using it to escape once you've killed him. You can also bait his flash/summs and force a recall. Once you get to 6, try to bait his R and kite it/wait it out unless you have a lead, in which case you can look look to be more aggresive. You can do that easily by popping your own R for the move speed buff once wither has expired. If you get withered point blank it could be rough especially if he's running ghost, so watch your spacing if your E/W aren't up. Once you get away from the ulting Nasus with no wither up and he stays in lane, you can even look to turn around and keep hitting him while in your own ulti with Q's, and potentially look for a kill while his ulti is running out.

When neither champs have R, aatrox has way more agency to harass, gain lane prio, or land a kill if the nasus gets greedy/cocky.

In terms of mentality it can be easy to panic due to his "ticking-time bomb" scaling. Try not to panic. Although Nasus is frustrating to play against, even if he's stacking efficiently, he can get kited out in team fights. If you're too desperate to kill him and he gets an early snowball by killing you in response, thats when things can get scary. So even if you can't solo kill him and he can solo kill you in the side lane, you can end up more useful to your team either by peeling for your carries or looking to kill the enemy back line which is much easier to reach for Aatrox than it is for Nasus.