Can my phone be repaired cheaply?

It is certainly not free, any more than your car company would repair your car for free if you drive it into a tree. Even if your iPhone was still under warranty, that doesn't cover user-caused damage. Apple does not repair iPhones, but will replace the entire iPhone for a flat fee; see the Repair Pricing section here:

http://www.apple.com/support/iphone/service/faq/

There are independent iPhone repair services that may be able to replace the screen for less. Do a web search for "iPhone repair". Note, though, that if you elect to have the iPhone serviced by an unauthorized shop, all further support from Apple will be voided.

Regards.

As part of Apple's rumored AppleCare policy changes, the company recently began replacing iPhone 5 screens instead of swapping out the broken units for refurbished devices, saving money for both Apple and its customers.

Priced at $149, the changes will also benefit consumers, as the previous price for a refurbished swap out was $229. Under the new plan, customers who didn't spring for the AppleCare+ warranty, which costs $99 up front and $49 per replacement, are now able to fix their devices at a more reasonable fee.

That's true, but the OP asked about a 3GS, not an iPhone 5, and I've seen nothing to indicate that they're repairing any problems with a 3GS.

Regards.

Missed that bit! 😟

http://www.iresq.com

http://www.drcellphone.com

Use Google and see if there are any repair shops in your area.