There is a shuttle bus between Domodedovo and Ryazan. As of November 2012 the bus was leaving Ryazan at 4am, 8am, 12noon, 4pm, and 8pm, transit time 3.5 hours (but remember the road situation and have at least two hours extra!), and the cost was 500 roubles, or about $17US. The number of the company called Uletny Marshrut is +7 800 555 1417 or +7 496 612-1728.
The bus will take you through Kolomna, which is “almost” a Gold Ring city half-way between Ryazan and Domodedovo. So you can combine getting yourself to/from Ryazan and seeing onion domes and tacky monuments in Kolomna.
In Ryazan buses leave from Michurinskaya Ploshad. In Domodedovo they are parked to your left as you exit the airport, close to the exit station.
THE ROUTE HAS RECENTLY BEEN SET UP AND CHANGES ARE LIKELY. CHECK DETAILS!