You can try the following but I suspect a hardware problem
- Reset the iOS device. Nothing will be lost
Reset iOS device: Hold down the On/Off button and the Home button at the same time for at
least ten seconds, until the Apple logo appears.
- Reset all settings
Go to Settings > General > Reset and tap Reset All Settings.
All your preferences and settings are reset. Information (such as contacts and calendars) and media (such as songs and videos) aren’t affected.
- Restore from backup. See:
iOS: How to back up
- Restore to factory settings/new iOS device.
If still problem, make an appointment at the Genius Bar of an Apple store since it appears you have a hardware problem.
Apple Retail Store - Genius Bar
Apple will exchange your iPod for a refurbished one for this price. They do not fix yours. I would say not worth it for a 1G
Apple - iPod Repair price
A third-party place like the following maybe less. Google for more. Also likely notr worth it.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Replace the screen yourself if you are up to it
iPod Touch Repair – iFixit