Ammunition doesn’t “expire” per se, but the gunpowder looses potency over time. The largest risk to shooting old ammunition isn’t a failure to fire, it’s the risk that you will actually fire the shot and it doesn’t have enough momentum to make it out the barrel. If this happens, the bullet becomes lodged in the barrel and becomes an obstruction for the next shot. This is a sure-fire way to destroy your firearm and possibly injure yourself in the process. If your shot ever sounds funny, safely unload the firearm and run a rod down the barrel to make sure the bullet isn’t stuck in there.
Yes you absolutely can, and we can help! Bullets follow very predictable trajectories governed by the laws of physics. The calculations which predict where a bullet will impact are no different for cannon balls or golf balls, and there are a lot of apps which can help you calculate this. As long as you know the bullet weight (typically measured in grains) and the velocity as it leaves the muzzle (usually written on the side of the box), you can determine where to sight-in at 25 yards so the round is zeroed exactly where you want it. For example, a standard 30-06 rifle sighted to 25 yards will also be dead nuts at 240 yards. Don’t believe it? Check out this article from American Hunter.