Omega Speedmasters do not have case numbers stamped into them like Rolex serial numbers. Instead they have individual Omega serial numbers stamped into the watch movement. This does mean you need to remove the caseback to be able to see the movement and to find out the Omega movement number. From this, however, you can begin to estimate the date of your Omega.
The only real way of being able to ascertain exactly when an Omega was issued is by contacting Omega directly through their website. They can issue you with an Extract from the Omega Archives for a small charge. This will give you specific information from the archives on how the watch left their production facility and the exact date of your Omega. This extract is simply that – it is not an Omega Certificate of Authenticity.
However for a good estimate of the year of your Omega watch you can date it by checking your movement number against this Omega Movement Number Database.