Linux kernel has had support for 64 bit time for years. On Debian, packages for the upcoming release were updated to 64 bit time earlier this year. I'm fairly sure the other distributions have done or are doing the same. So basically you now have 2 years to upgrade your OS and to pester the vendors of commercial software to do the same.
Like someone else said, it will be 2 very busy years, but we can survive this.
Whenever you get this kind of thoughts, take a moment to also think about the maths behind your CT and MRI scans, which originated from early radio astronomy. Alas, I don't have a source for this other than it was said by an astronomy professor during a lesson for an exam I never even attempted.
You don't need a desktop for a server. A local TTY allows the same stuff as connecting over SSH but you can do whatever you need to bring up the network. The drawback is having to go to the data centre...
Would be fun to put that in a luggage and ask someone to help lifting it to the car