Google Transit doesn't have the Bee-Line system registered. You'll have to go directly to the Bee-Line website here
The #13 is scheduled to take about 25 minutes to get to the White Plains Transit Center, but I'd factor in some extra time for traffic, so make it say, 35 minutes. You don't have to walk all the way to Downtown Port Chester for the bus. You can take the #61 bus over there. (See the schedules at the link above).
For SUNY Purchase, the roads don't look particularly bike-friendly, but they don't look too bad. If you decide you can't do it, you can take the #13 bus to the #12 bus at Purchase Street.