Well, the old joke was that you had to saw the gun in half, and count the rings, but.......
Serious answer- there is no one way to determine that age of a firearm. It varies wildly from one maker to another.
For Ruger and Browning, go to their website, and look under Customer Service, Historical Data- and they list serial number ranges from one year to another.
For Marlin Lever action rifles, the letter prefix to the SN indicates year, until late model guns changed to a different pattern in the serial number.
Remington- it was a 2 or 3 letter code stamped on the barrel, would tell you month and year.
Winchester- 2 different goups of experts argue over years for different serial numbers, but the Cody Museum in Wyoming has the records from Winchester.
For many guns, closest you can get is a date RANGE. I collect early Remington and Mossberg 22s. They did not HAVE serial numbers. So that goes to "What years was a Mossington Mdl Q36 made?"
There are very few references for H&R. Mr. Bill Goforth, the most knowledgable man I have ever encountered on Iver Johnson and H&Rs wrote some of those, but Bill passed away last year.
H&R made (for the most part) good, sound, durable utility grade shotguns- the gun that belonged to the working stiff- the farmer, shop keeper, mechanic. They made them from 1874 to 1986, when the company closed. The name was revived for a few years, but I think THAT has gone away now. One of us here may have a reference that lists your shotgun, I regret I do not. If yours does not have a serial number, it is from before 1968 (not required on rifles and shotguns by law until then.)
PS- my first gun was given to me on the day I was born by my grandad- a H&R .410 single shot.