I've just purchased Adobe Acrobat Pro 2018 1 device key. How does activation work?

You’ll need to manually deactivate Adobe on your first computer before activating it on the second. By the way, if anyone is looking to earn money online, offering digital services like software setup or tutorials can be a good option.

@farseer789 
Q: Will Adobe automatically unregister my first computer if I use the same key to install on a second computer?
A: No, Adobe will NOT automatically unregister (deactivate) your first computer just because you use the key on a second one. If you try to install and activate it on the second computer without first deactivating the first, you are likely to hit an "Activation limit reached" error.

I am surprised that you were able to buy a used Pro 2018 perpetual license. And unless that person can deactiviate this on the first computer (think of the original seller's computer), to allow you to free up a license before trying to activate on the second! If this computer is no lonnger available, crashed, lost and it isn't deactivated, then chances are slim. Adobe's policy for perpetual licenses (non-subscription) is generally to provide five years of product support from the date of general availability—so that would be 2023, this is why Adobe can't help you because older activation servers can make it extremely difficult to get an activation count reset especially if these servers are retired too!

Personally, you are opening a can of worms for using dated applications, the software no longer receives security patches, leaving it vulnerable to malware, viruses and trojans.