The short answer is yes, he eventually gets Aria's blade in his throat.
The long answer is that Grave is stupidly powerful so not any time soon. After this first encounter Aria survives not because she bested him, but because she recognized she was completely outmatched and uses her wit and cunning to fake her death. Even Grave himself has a feeling she survived and remarks that if she actually did, then she'd be an interesting opponent. This encounter has a heavy mark on both their lives, Aria because Grave becomes her new goal, as in she needs to prepare for when she needs to confront and kill him, and Grave because taking out Aria was a rouge action and he is actually being hunted by the black ops main branch after. He goes independent.
After some massive power leveling by Aria, the rainbow swords (The adventure party that our "Oji-san" is part of) is contracted to hunt and kill Grave. Both because he's a rouge agent and intelligence suggest he's going to assassinate a duke. The Duke is hunting a super powerful beast that basically a panther and Grave plans on assassinating the Duke at this time and prepares a number of things. Aria and co "corner" him only for him to sic the panther on the party. Aria leads the beast away so that her party can confront Grave and in that time earns the respect of the beast by showing that she will fucking kill it dead after besting it. The beast asks for a name to which Aria names it Nero. Riding Nero back into the battle Aria directs it to attack Grave. She takes one of his arm and his favorite weapon, but Grave escapes by being a crafty bastard.
The final encounter is when some demon activity started popping up and it turns out their co-ordination is because Grave was planting intel for them. Aria confronts him and we see that Grave replaced his arm with a magical mechanical one and during the fight reveals that he actually copied some of Aria's equipment and skills. Aria is once again outmatched but the gap wasn't as big as before. Grave offers to take Aria as his underling to which Aria replies with a Gigachad "no". Fight ensues. Aria plotted her battle by tossing a super poison that melts people's lungs and reveals to Grave that she took the antidote so she will last longer and Grave has 2 options, try to escape and be hunted by Aria in a very weaken state, or confront and kill Aria now while he still has strength. Grave is enraged that Aria would just blindly throw away her life like a dog and continues the fight with Aria but Aria, just a bit quicker stabs Grave in the throat and ends him. Aria remarks how she was not throwing her life away, but this is now she grows and that she will never see eye to eye with Grave's ideals.
