Based on function:
* Faucet (more general term)
* Valve (a device controlling flow)
* Tap (British English for faucet)
* Closure (general term for something that stops flow)
Based on opposite action:
* Stopper
* Plug
* Clog
Based on opposite location:
* Reservoir (where the water originates)
* Source
* Tank
The best antonym depends on the specific context in which you are using the word "spigot."