The old mc servers we had are 1.12, if you wish to open a 1.13 survival server, you're gonna have to regenerate the world either ways, that means the buildings wont stay unless you copy and paste them between the old and new worlds using WorldEdit, and you'll have to track their locations too inside the world, it's possible, but I personally prefer everyone starts fresh. What you can do is, simply run the server as it is, make sure it's working well (you'll have to change the IP in some plugins' configs and some databases like in Logblock). Then if you want to save old buildings from that world, locate them by going to each faction home (/f home Animosity) for example, or a player's own homes (/home velaxers:), then save schematics of them using WorldEdit. Now for the updating part, shut down the server and keep a backup at this point. Start with updating the spigot.jar to the latest stable 1.13, as well as updating every plugin in the Plugins folder, just search up each plugin name and use the Spigot or Bukkit websites to download the latest version of each plugin (only replaces the .jar files of the plugins, don't delete the folders unless you want to reset all the configurations that were done before). Almost every plugin has a folder and configurations for it inside the plugins folder, you might need to check each of them to set up the server the way you want. I've already set most things before, but by updating the plugins there will be new options and features. Also, we used to use Factions plugin for land protection ( "factions" plugin does not mean a factions server, it's a land protection plugin after all, to avoid griefers), a better alternative is
GriefPreventation. Right before the release, if you chose to let players start fresh, reset their data.
Another way you can use is ignore all that shit, launch a brand new 1.13 server, and search up the names of each plugin from the old server and learn more about them, to save yourself time finding what plugins you need, then set them up from scratch.
Excited to see the server up, goodluck!