Can my 2013 iMac be used as a display for a new Mac mini?

Your mini would be too new to use Target Display Mode:

Use your iMac as a display with target display mode - Apple Support 

There are third-party solutions such as Luna Display, but they are only suited for uses where there will not be fast motion and where lag time is not an issue, such as for tool palettes for an app. I think you'd be happiest just buying a new monitor.

Regards.

No, not using Target Display Mode.

You will need an Aftermarket or Studio Display to Setup the new Mac mini and for the Primary Display. Then you can use the iMac as a secondary display with screen sharing or third party software if you wish.