First, I'm sorry to hear about your grandfather. It's never an easy thing for any of us to lose a loved family member.
As to the rifle, give the first part of the serial number (X out the last 2 digits), and a couple of folks here can tell you exactly what year it was mfr'd.
As to the model, here's what I know (with referencing Murray's).
Model sounds like the Savage 99 Brush Gun. Has a straight stock, 22" barrel, sling studs, came with a recoil pad. Made from 1977 to 1980, and only offered in 358 from 1977 to 1979. The 375 was added in 1980. Has a tang safety, rotary magazine with cartridge counter, pin-type cocking indicator on the top. Forearm should have finger grooves on it.
If you stay around and soak up some 99 lore from us, you'll hear lots of references to pre-mils and post-mils. In 1960, the factory moved and the serial number range went above 1 million. And cost cutting measures were introduced and so a lot of the old style quality and class disappeared forever. Thus, rifles made in 1959 and earlier are pre-mils, and 1960 and later are post-mils. So the post-mils are generally looked down on by collectors.
The 358 and 375 brush guns are not looked down on. Folks may grump that they don't have the same quality as the old rifles, but they are
very desirable rifles to own. Add into this the origin of your rifle, and it's a priceless treasure. So when folks rant about how bad post-mils are, don't let it get to you.
The Savage 99 series started as the Savage 1895 which was made from 1895 to 1899, then the Savage 1899 which ran from 1899 until early 1920's, then the Model 99's were made up through the 80's, and a few later. Browse on over to
www.savagela.org and you can view pictures of most of the various models made during the years,a nd we'd love to see a picture of yours if you get around to it.
Welcome to the forum!