Here is an update on taking a taxi to and from the airport to downtown Asuncion.
The cost of a taxi from the airport to downtown Asuncion was 120,000 Guarani in November 2017. This is a set fee. I paid in Guarani, but I suspect they may accept USD. The airport is small. There doesn't appear to be an official taxi stand. If you exit from the left and head slightly to the right there is sort of a booth where they showed me a document with rates.
The hotel called a cab for me to return to the airport. I left at about 3 a.m. The fee was 100,000 Guarani from downtown Asuncion. It appears that this is the set amount regardless of when you leave.