You need it before your trip.The airline agent will not issue you a boarding pass if you do not have a valid Visa stamp. However you can not apply for it until you have purchased your ticket as it is a condition to provide proof of travel plans( specific dates for which you are already ticketed). Once you buy your airline tix, you may then apply for a Visa. You must use the visa for the intial visit within 90 days of issuance. Thereafter it is good for multiple reentries of uptp 90 days for five years.

BTW, the process is not as daunting or complex as it may seem,So long as you are not holding employment on your tourist visas, they are happy to have you and your Amerrican Dollars visiting.

Finally, welcome top FT and congrats on your passport!

Rio is an amazing city, You'll love it