Select a Linux distribution "for enterprise use", install just the packages needed for its job (extra packages mean extra vulnerable surface), don't install any "unofficial packages" unless strictly required (and then only carefully considered ones for stability and upstream responsiveness to security problems, and track record/commitment to not trampling on the distribution's packages), configure with care (local firewall rules, any local configuration). Create local accounts as needed, give them secure passwords. Use
root as little as possible on the machine. Set up passwordless SSH for selected users. Don't follow the many suggestions on the 'net to "just turn off SELinux", SELinux is able to provide much added security. Work with it, if a package doesn't work with SElinux, it is broken and shouldn't be used.
Subscribe to the distribution's package update announcements, update at least the security updates ASAP (not "once a month", at least check daily). Look at LWN's security page regularly.
Whatever services the machine will provide, select the packages to do so with care among what your distribution offers. It will probably be that the overall best is harder to use... some ideas on selecting/evaluating software are here.
Look around for "best practices" for the services to be offered, in particular configuration suggestions for what you install. Check if the online documentation is complete, clear, and searchable. See if there is a way to report bugs, check a random selection of bugs to see how responsive they are. Look for online question and answer sites, FAQs, mailing lists (if you run into trouble, you want to know who you can ask).
Design a comprehensive backup scheme, sooner or later you will have to fall back on it. Document the installation (or even better, set up automated installation, and keep it up to date), something untimely might happen to the machine. Make sure you have "old" backups too (some miscreant might take over the machine, and you find out a few months later...).
Consider hiring a few geek specialized in what you want to set up, just make sure to (a) keep them close at hand, and (b) you understand enough of the setup to be able to muddle through for most cases, and (c) make sure you pass the bus test (i.e., have others around that can take over in case you get run over by a bus).
Set up a machine with the same operating system for personal use, it is very useful in case of crisis to be intimately familiar with the ailing system (or even have a spare on hand ;-). This will be a part-time job, as in "it takes only 95% of my time", unless you take the time to organize the tasks so to minimize the unexepected (or you end up being the fire brigade being called 24/7).
Personally I'd use either Red Hat Enterprise Linux or a clone like CentOS, perhaps supplemented with EPEL. But that's just me, Red Hat user from around '95, lately Fedora fan.