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There's no great way to improve the quality of an already made video. You can try using the Sharpen effect in AE but that won't really do much.
In fairness to the tutorial creator, looks like it was made in 2014 and YouTube compressed it down to 360p. You could always comment to the creator and ask if they still have the original.
The creator does explain everything he's doing, but of course the interface has changed since then too.
This similar video is on Adobe's help site: