At the risk of getting a bit macabre, let's have a brief conversation about how guns kill people. There are basically three mechanisms by which this occurs (at least in the short term). One, the bullet can destroy the central nervous system, either by striking a critical part of the brain or by severing the spine. This is basically instantaneous and by far the quickest way. Two, it can shut down the cardiovascular system by destroying the heart, which will generally take somewhere between a few seconds and a minute. Or three, it can bleed the person out by either rupturing a major blood-bearing organ or severing a significant vein or artery. This can take anywhere from a minute or two to several hours, depending on the severity of the wound.

We'll use your example of the rifle. Now, high velocity rifles are good at inflicting all three types of wounds, but they're particularly effective as compared to other projectile weapons at the latter. Because the bullet arrives at such a fast speed - say, over 2,000 feet per second - it generates something called hydrostatic shock and a temporary stretch cavity. In layman's terms, the bullet is moving so fast that it creates a sort of shockwave that disrupts nearby tissues, so it can rupture an organ or a blood vessel that it passes within a few inches of. This means that rifles are much better at bleeding people out, even if they don't hit the brain/spine/heart.

Arrows are pitifully slow by comparison. Whereas an M4 carbine will generate upwards of 2,800 feet per second at the muzzle, and a musket will do around ~1500 FPS, an arrow from a powerful bow will top out at about 150 FPS. This low speed bears on the discussion at hand in two ways: one, it means that the arrow is a less efficient penetrator and may struggle to pierce deeply enough to reach vital organs; and two, it completely rules out hydrostatic shock. Shooting someone with an arrow is essentially like poking him with a pencil - the size of the projectile determines the size of the wound channel and the damage done.

If you're looking for some kind of an average time until death, those figures just don't exist. But yes, we can safely say that bows are significantly less efficient killers than firearms. How much so is dependent on circumstances. An arrow buried in the brain stem will produce virtually instantaneous death; arrows which merely lodge in soft tissue but fail to pierce an organ or sever a major blood vessel may very well not be fatal at all, even if a person is struck with several of them (or it may kill him a week later from infection). This is true for bullets as well; it's just that the bullet has a bit more margin for error.