What is the manufacturing date of Remington 22 long breech loader?

Sounds like you are referring to a Rolling Block Rifle, would need to identify which one.
Most common would be either a Model 4 or Model 6, a Model 6 is basically a cheaper version of the Model 4 which was made to compete on price at that time.
The Model 4 was made from 1890-1933 and the Model 6 from 1901-1933