you can use the same schematic and probably the same code. But look at the the 16F628 Specs... it have an internal USART. More reliable than a software solution and produce a tighter code. HSERIN, HSEROUT that's it.

About skipping the external OSC... it could work good or not. Usually for serial comm i prefer to use a crystal for a better stability upon the temperature variation. The internal OSC is still a RC circuit... But try it before at slow baudrate and it could work great.