I have a Samsung S4 mini and a Nexus 10 tablet without a sim card. Is it possible to use the phone as the tablet's internet connection?

Yes. set-up the phone as a wifi hotspot.

http://devicehelp.optus.com.au/web/samsung-galaxy-s4-mini/connectivity/setting-up-and-using-my-mobile-phone-as-a-wi-fi-hotspot/

Then search for wifi on the tablet and connect.

Heres one site that explains it. There are many more out there.

http://www.tech2touch.com/how-to-use-samsung-galaxy-phone-as-modem-to-connect-to-internet-from-pc/