You would be surprised at just how many cabin crew around the world are on some type of medication for depression and/or anxiety.
I'm surprised at the very naive comments in this thread. Someone with anxiety/depression can deal with emergency situations. I'd be more worried about the crew with poor EP knowledge than someone that suffers from anxiety.
Anyway, to answer the thread starters question's.
Each airline has it's own policy on medication. Some allow SSRIs etc but must be disclosed in case of drug testing. You will be drug tested at the medical stage, and may raise questions by the doctor if you haven't declared anxiety as a pre-existing condition.
You asked about how long the drug may be present in your urine after you stop taking it. I'm assuming you are already aware that you shouldn't suddenly stop medication like effexor as the side effects will be significant. But, all traces should be gone within 3 days.
Good luck!