Well, how would you operationally define updated? Would you, as a traveller, be satisfied if it's no more than two days out of date?
How long does it take to check and update the information for one country? Let's call it 90 seconds when you have the routine in. (Maybe every tenth time you check you have to actually change something in your database and that takes 10 minutes, the rest of the time you change nothing and that takes half a minute.)
There are about 200 countries in the world, so updating them all takes on average five hours. If you only need to do that every other day, that's quite literally 10 % of your time to do manually.
Now it could probably be optimised -- some countries might not be as popular destinations, and others might not change their rules as often. This is data you get for free from the effort of maintaining it. You can use that to adjust frequencies and I'm sure get it down to just 3 % of your time or less. That's while still doing it manually!
Then if there are some places that are really popular or change really often, you can start automating the updates for those countries. Since they are the ones you'd spend the most time on updating manually otherwise, you can probably get it down to less than 1 % of your time.
But never forget to start by doing things that don't scale. You can get very far with a good manual process. Automate only when you have a good manual process and you've driven the last inefficiency out of it.