connect airport time capsule to my network

The Time Capsule WAN port needs to be connected to your router.

Ideally your Mac will have its own, separate ethernet connection to your router; but if necessary you can connect your Mac to one of the LAN ports on the Time Capsule. (And I believe that in order to do that the TC has to be set to Bridge mode.)

Here is the TC Setup Guide -> https://content.abt.com/documents/17962/Apple_TimeCapsule_SetupGuide.pdf Note that you also have to use Airport Utility to set up & configure your Time Capsule.

You do know that the TC was discontinued 6 years ago ...

mac mini only has one ethernet which I use to plug in to my router

Connect an Ethernet cable from your router to the "O" port on the Time Capsule

Connect your Mac mini to one of the three LAN <--> ports on the Time Capsule. Choose any of the 3 ports that you wish....since they are all the same