you need to find the serial number, then do a web. search for the gun maker, or others who have done the homework for you.
You can look that up here http://oldguns.net/sn_php/marlinlookup.php
Contact marlin customer service through their website.
The Wikipedia site for marlin 336 rifles includes a table showing how to use the serial number to find date of manufacture.
Try this link--- http://oldguns.net/sn_php/marlinlookup.php
There is a web site called "marlin manufacture dates", but it will only give you the year of manufacture, not an actual month and day. Seeings as you already know that it is an 1883 rifle, then you probably know as much as you are going to find out about the rifle.
While I cannot give you a specific year of Manufacture,I can say that it was made after 1906.You can find the exact year of manufacture by contacting the Cody Firearms Museum in Cody WY.There is a 60 dollar fee for a letter stating the date of manufacture and when the rifle was shipped.
For Marlin manufacture dates head to the site linked below. (see related link at the bottom)
If you provide the serial number in your question then I can give you the date of manufacture.
By it's serial number.Go to proofhouse.com to find the date of manufacture.
Starting in 1973, the year of manufacture can be determined by subtracting the first two digits of the serial number from 100: Example: SN 2512345 would have been made in 1975 [100 - 25 = 75].
If it has the 17 digit vin, the 10th digit is the code date of manufacture.
If you would post the serial number and model number,I can give you a date of manufacture.