Hi Vassily,
Firstly is it possible to do so, secondly will it carry over my data?
A: Yes, it is possible to change an existing managed disk from 128 GiB Premium SSD to 512 GiB Standard SSD. Yes, it will carry over your data.
When selecting the plan and storage I want I do receive this warning. "Changes to the disk size can be made only when the disk is unattached or the managing virtual machine(s) are deallocated." To my understanding all that means is the Virtual machine needs to be shut down and and stopped Via Azure portal.
A: The VM needs to be either Deallocated or the disk needs to be unattached. For OS disk, you would Deallocate the VM. For Data disk, if no applications are using it you could Detach it from the VM, make the change, then re-Attach it to the VM. Or you could Deallocate the VM for Data disk, your choice.
What I recommend is you first Deallocate the VM (click Stop button in portal), wait for it to finish deallocating, take snapshot of the disk (for backup), change the size and type of disk, Start the VM, connect to VM (via Remote Desktop or ssh), and finally resize/create partition(s) as needed.
After several days or whenever you feel comfortable, delete the snapshot.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP