I found out that neither if-statement nor case statements are allowed in FQL.
I found a list of available functions, operators etc. at http://steve.veerman.ca/2012/facebook-api-fql-tutorial/
My solution was to delete the following from the query:
AND
start_time >= now()
I also added now()
to the select statement, to get the current time (despite that I could do that in PHP).
I would then get all the all events out, despite how relevant they were. I would them sort them with a if-statements and compare the start_time and the end_time, if it had a end_time.