Whatever you do, DON'T try to sand the wood until it lightens. It's not possible - the stains will go deeper into the pores of the walnut than you'd want to sand. While sanding it might LOOK lighter but that's just the sawdust you make and as soon as you go to apply a finish your stains will still be there but you'll have a badly wallowed out stock that will never look good again.
I'd try a leaching/bleaching product as rcmodel described in post #2, then let the wood stand for several days to dry out. You want it as dry as possible before beginning to put any color or finish to it.
Once really dry I'd set to trying to blend the color overall using spirit or alcohol based stains. If you use oil based stains they'll permeate the wood and you won't be able to make much adjustment from that point on.
Try a light color and rub the wood with a soft cloth to try to get an idea what it'll look like.
If it's too light to blend the oil stains and look natural then move to a darker stain.
Best to stay with brown colors - not the heavily red 'military' stains that are everywhere. Brownells has a 4 bottle kit of spirit stains in different shades that will have something that's close to right but it's important to do test patches ON THE ACTUAL WOOD you're working with. Walnut is weird in the variety of ways it takes color and it's like no two pieces will look the same when colored with the same stain. I guess just saying "Walnut" isn't good enough to describe the wood.
Anyway the blending plan as described by ralfus is the way to go and keep in mind that it's a classic old rifle. It would not be appropriate to try to make it look like a new Browning made in Japan.
Most of the model 99's came with a light varnish or shellac that's not easy to duplicate with new products. A slow hand rubbed finish using boiled linseed oil finish could look good (not much is finer, IMO) but it's got to be done right and that means a lot of work - a lot of rubbing with bare palms and enough pressure to make it seem like there's no way you're not raising blisters. The heat from rubbing sets the oil and gets it to dry so it won't bleed out on every hot day. A lot of people just wipe on some oil and call it a finish but all they've really got is some oily wood.