It is unusual vs. rare
1930, check the date code stamped on the barrel. Remington Society of America, Manufactured Date link has the details.
A J sounds like this is a model 4 rolling block??? Also there's no other letters.It that's a 0 it's 1928 if not it's 1929 right on the boarder. IF it's a model 4.Also if a model 4 it will be date code stamped indicating month/year on the left side of barrel in front of the receiver, you can look up diagram and codes on the Remington Society of America
Remington never made the Model 4 in 270 Winchester.
I only have the serial list to 1931 and it's after that. It will however have a date code stamped on the barrel see related links for a page with details.
Remington did not make the Model 4 in 30/06. Value as a shooter if cleared by a gunsmith as safe. 100-300 USD
You will have to contact Remington.
100-400 USD
Impossible to answer without the serial number.
The Remington model 4 rolling block rifle was made between 1890-1933.If you have a model 4S military rifle,then these were made by Remington from 1913-1923.your rifle has either a 22.5in barrel,or a 24in barrel.These were chambered in .22,.25 Stevens,or 32 caliber.
The Remington Model 4 was made from 1890-1933. Check the barrel on the left side for letter markings (2 letters, just forward of the receiver) that are the code to date barrel was made. You may also check with Remington.
Your Remington model No.4 rolling block single shot rifle will bring between 250-500 dollars for a model showing between 10%-60% of its original finish,and a good bore.This rifle was made from 1890-1933.production totals were 350,000 made during this time span.Yours was made toward the end of production as per your serial number.Almost certainly last year production being 1921 or later it will be date code stamped on the barrel, see related links for information on the Remington Society of America.
My research indicates about $300.00 in good condition, and that is for a model 4 in .32 long caliber.