How do I tell what width chain my used chainring needs?
If it's 1/8" then a 3/32" chain will have to be jammed on to mesh, or not fit at all. The simple thing to do if possible is take a chain from a derailleur bike or even a scrap and physically try it. 

1/8" chains and cogs/freewheels tend to go longer and on a singlespeed application, there is no downside (other than weight) to running them alongside a 3/32" ring, so you could also just get 1/8" for the rest of it regardless of what the ring is and call it good.