How do you figure the united states railroad annuity start date?

What month/year  did your retirement payments start ?  That is what you need to enter ... there is nothing to compute it is just an actual date. 

The "annuity starting date" is either the first day of the period for which you receive payment under the contract OR the date upon which the obligation under the contract becomes fixed, whichever comes later.

For example: On January 1, you finished paying all payments required under an annuity contract providing for monthly payments starting on August 1 for the period beginning July 1. The annuity starting date is July 1, which is the first day of the period for which you will receive payment.

Any chance you can expand on your answer? In other words, is the start date this calendar year or since the very first payment was received 20+ years ago?

The start date is the date you received the very first payment.