Is this an older take down model or is it a solid frame? The savage 99 is really easy to clean without taking it all apart. If the butt stock is removed (two screws in the plate then the long screw in the hole) and the butt stock is removed. Most of the internals are visible and can be cleaned. You have to be real careful removing the butt stock as that is how you can crack the stock. It has to removed straight back without and wiggling back and forth. To remove the rotor and clean it is a little more complicated. I have done it, but it is not something I would recommend for a novice. If you want it cleaner a pretty good job can be done with a q-tip through the hole. If it is a take down and you are trying to remove the barrel make sure that the bolt is in the open position. When reinstalling the butt stock don't over tighten it. That can also result in the tang cracking. Tighten it snug like you would an oil filter on a vehicle, not zinging tight.
Good luck