Can you connect a 4th generation iPad to a printer with a cable?

You can't connect ANY generation iPad to a printer with a cable. Printing is done exclusively by wifi on ALL ipads.

And more specifically ringing over Wi-Fi uses AirPrint. The printer must be compatible with AirPrint. Not all printers are.

http://support.apple.com/kb/ht4356

If you don't have a wifi printer, you can get an app off iTunes that will connect to your Printer via your pc (can't print from all apps though)