How long do sperm live in a sterile cup?

Less than 90 minutes. Usually you want to get the sperm out of the semen by 60 minutes, but as long as the temperature is moderate, sperm can survive 60-90 minutes. Once taken out of the semen, sperm can survive a long time in media. In the cervix or uterus, sperm last 3-5 days while in the vagina about 1 hour!
30-45 minutes. Sperm can survive longer than 30-45 minutes but to remain viable for pregnancy the sperm really should be processed in that time frame. Sperm not processed within 45 minutes often lose their ability to move around and get to an egg.