If I sign up for the monthly pro version for only one month, can I still get the one year membership at the discounted rate after the month is over?

Yes, anytime you want to purchase a yearly subscription you still get the discounted price. If you switch from monthly to yearly on iOS, your yearly subscription will start after the monthly subscription ends. If you switch from monthly to yearly on Android, your yearly subscription will start immediately and you will be refunded a prorated amount based on how much time is left in the month.