Keep Mac mini on Wi-Fi (for screen-sharing), but disable internet access... How?
There are a couple of things you can do, but by far the easiest is to just manually configure your IP address (don't use DHCP) and leave the router field blank.  See below...



Before you select "Manually" in the drop down, take note of the IP address that was assigned to it by DHCP.  Then when you select "Manually" just re-enter that same IP address.

With the Router field empty, your Mac Mini will not know how to get out to the Internet.

Update:  As pointed out by user3439894 in the comments, reusing the IP address from the DHCP pool could pose a problem.  To avoid this, assign it an address outside the pool.

Most routers will have a starting address (Netgear's typically start their addressing at 192.168.1.100 and assign 50 or 100 address).  So you either have addresses up to 192.168.1.49 or .199.  To make sure you don't cause conflict, just assign it an address either higher than or lower than the pool (i.e. 192.168.1.30 or 192.168.1.210)