You can call Remington at 1-800-243-9700 and go thru the menu selection. One of the menu options is to request information on your Remington based off the serial number. You read off your serial number and the guy looks up in the database the year the gun was made and what the original barrel was for that gun (if gun had barrel options like shotguns do).
Impossible to tell without the serial number and all markings on the weapon.
1990
http://en.wikipedia.org/wiki/Marlin_Model_336
35
They only made them from 32-35 and they are date code stamped, see related links for information on the Remington Society of America.
The letter S was assigned as a prefix letter to the serial number from June of 1958 thru 1959.
1950.
1949
The 141's are date code stamped on the barrel, the location diagram and date codes are on the Remington Society of America.
No 35 Remington fired only 35 rem. The 35 Remington is a good cartridge, it's a little more difficult to find but it is still being made.
http://en.wikipedia.org/wiki/Marlin_Model_336
Depending on condition, value may be $300-450. Age can be determined from a 2 or 3 LETTER code on the barrel- left side, just in front of the reciever. The Remington Collector's website has a chart showing month/year using the letter codes. Google search Remington BLACKPOWDERX for the chart.