Well depending on how they have made the original anime in the first place, and how much they kept, it range from "hard" to "just redraw everything, it will be easier".
The first thing to know is that production studio usually don't keep files / puppets after delivery.
They usually keep some assets if there is a chance that a new season will be bought, but that's it.
Now, to "fix up" like your video shows, they basically kept the original data, and edited the images afterwards.
answer was correct.
Don't think that VLC filter can fix up a fucked up face tho, it will only remove some noise, and sometime improve colors.
To really upscale and clean an image, you need an AI or some humans to redraw everything frame by frame (except if they have access to the original animation files, then they can only redraw what's important, and the background and such will stay the same.)
Have a look at Waifu2x. I know it's old, but it use AI to upscale images, and take a good amount of time using only CPU (remember, you have 12 to 24 images to upscale for each second of video).