Apple will exchange your iPod for a refurbished one for $199 if youhave a 64 GB one and $99 for the others. They do not fix yours.
Apple - iPod Repair price
A third-party place like the following maybe less. Google for more.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Replace the screen yourself if you are up to it
iPod Touch Repair – iFixit