Will an 11-speed SRAM shifter work on an Alfine 11? I'm building a bike, got an Alfine hub for cheap, but the shifter feels like garbage. There're lots of new 11 speed shifters out this year, like the SRAM X1, do they work with the Alfine?
No. SRAM shifters (and regular Shimano shifters) don't have the right cable pull to deal with an Alfine hub. 

You need the Alfine shifters or ones marked to be compatible with Alfine (Jtek barends and Versa brifters are two of the 3rd party options for road bikes, for example).