The "Hint 5" on this problem is
going to be outdated in three days already outdated.
Should I edit the hint when it gets outdated?
Are similar hints that highly depends on time allowed?
Hints that are easily outdated can be a problem, but probably not in this particular question. If the hint referred to knowledge that's likely to change and be unavailable to future solvers, that's clearly something that should be avoided as it could possibly make the hint unusable and misleading. Let's say for example that you refer to a particular Stack Exchange user's profile picture. If the user changes their profile picture, there is no indication to future solvers of that particular puzzle that the hint is no longer valid.
However, in the linked question the hint reads
Where was the last Winter Olympiad held?
Questions come with dates and timestamps, and past Olympic hosts are public knowledge, so even if someone comes to this puzzle years from now they can still figure out what is meant by the hint.