I don't think that it's actually the case that spider plants (Chlorophytum comosum) prefer to be rootbound. In fact, I take the sort of extreme position that no plants ever actually prefer to be rootbound. (My theory is that for those plants that people say like to be rootbound, people are confusing cause and effect: they see a healthy, older plant that is also rootbound and assume that the plant is healthy because it's rootbound, where I think that it's rootbound because it's older and healthy.)

Though this is kind of beside the point. Move it up into a pot that's two to four inches larger in diameter (two is probably better for the plant, but you'll have to re-repot sooner). Use an actual potting soil, not topsoil or garden soil. Something that's mostly composted bark is good.

The brown tips are most likely from mineral buildup in the soil (whatever soil is left in the pot, anyway: I'd be surprised if there's much in there). This is especially likely to be the cause if you've been:

1) letting the plant stand in water because it wasn't soaking any up when you poured it through the first time.
2) fertilizing regularly.
3) watering frequently in small amounts, rather than watering heavily and then dumping the drainage within fifteen(-ish) minutes.
4) any combination of 1), 2), and 3).

I've always just watered til the soil was moist and the excess was starting to drain out the bottom (there's a drainage hole)
I've actually never used a fertilizer in the soil because it's done so well for years without being fertilized, Do spider plants need very much? If so, what's a good one to use?
I'll definitely re-pot it if it will help the plant though. When I do that will I need to clean off the roots of all the left over soil, or is just getting most of it off okay? Should I clean the roots in transition, or will that 'stress the plant' so to speak?

Thanks!

More thorough watering (like putting the plant in the bathtub and turning the shower on until it's completely saturated with water) will keep the mineral buildup down, which should keep it from developing the brown tips.

Not fertilizing much is definitely better; they don't need a lot, and too much will, again, lead to the brown-tip thing.

I don't think you probably need to worry about cleaning the roots at all when you repot. You shouldn't have to do much more than put a little dirt in the new pot, lift the plant out of the old pot, set it in the new pot, and then fill in the space around the root ball with new dirt. Spider plant roots aren't what you'd call delicate or fragile.

Repotting is usually encouraged to be done when the plant is actively growing. This speaks for when the sun is high--in spring and summer, not in the winter.
Since it is only September --going into October, I think its safe to assume the plant can take a re-potting at this time without undue concern for the light levels.

Brown leaves usually indicates a water problem....or maybe a smoisture problem. Since the plant is mostly seen hanging up, and sometimes not with a saucer that catches the drainage properly, they can sometimes get ahead of watering and dry out more than they should.
Sometimes too, when soil dries out, the soil in the pot shrinks from its sides and when watered, it goes right down the sides and out the bottom, never actually making contact with the roots.
A once in a blue moon leaching by letting it sit in water, submerged, can give assurances it is indeed watered.
But this is used uncommonly.

Possibly too the temperature of the water can cause browning of the leaves..or the tips of.
Let the water sit overnight to gain room temperature and lose some of its fluoride.

If not hanging up at this time, you might do this so that it can get as much light as possible now that the sun is going down in intensity. Being in a room with light colored walls too can offer the plant increased amount of light. The eastern exposure will do for a time but eventually should be moved to more southern or western light.