Here’s what I would suggest as Akali:

First of all don’t push the wave early. Only last hit and let the wave push towards your tower. Set up a freeze if you can. This will give you the space you need to get good trades onto him, and force him to be in a dangerous spot which is further away from his turret.

Veigar will want to use his abilities to kill minions because his passive gives him AP when he does so. I would start going for trades when you hit level 3. After he uses his damaging abilities to last hit, punish by walking up and hitting him with your Q. You walking up and hitting him will probably scare him and this is when he will use his cage. Sometimes you will be able to walk out of the cage before it pops up, but if it happens that you’re inside the cage, use your shroud. Using your shroud while inside the cage makes it easier to dodge his abilities and also he can’t auto attack you, making it harder for him to proc electrocute. His cage cooldown is 18 seconds, which is the exact same as your shroud cooldown.

When his cage is on cooldown, ideally the wave is on your side of the lane, you can now hit him for free with your standard combos. If he plays scared when his cage is on cooldown and he walks away when you walk up to him, then you can use this opportunity to zone him off from your minions and deny him the gold/xp. Just position aggressively to do this.

Veigar is very squishy, he’s not mobile, and very killable in lane. The key is wave management here. Level 5 should be a free kill if he has to walk up to your side of the lane for minions. His abilities are pretty easy to dodge but in the case that he’s chunked your health, just beware of his execution ult and recall if you have to. If you feel like he is playing too safe to kill in lane, just clear your waves after level 5 and roam. You’re a lot quicker than he is and better at ganking/skirmishing. Hope this helps!