"First, catch your hare …"
First, you have to define what "network is ready" means. It's not simple, and your definition may be quite different to someone else's. Your question, of course, provides zero inkling of what your particular idea of "network is ready" happens to be.
When you've defined that, the rest is a simple matter of writing a unit file to replace your old System V shell script, which is a good idea anyway, and adding the necessary After=whatever (and probably Wants=whatever as well) to the unit. The hard part is the whatever, which depends from what you think "network is up" is.
When you've written your unit file, you might care to donate it to the world, making it one less unit file for the world to write.