Is it possible to program a AT89S51 chip with a USB to TTL(UART) converter cable or Arduino?
The adapter you provide in the schematic requires bit-banging the protocol on the serial port pins. It is doable on computers with motherboard serial port.

However, when it comes to USB-RS232 adapters, most can only send data via the Rx-Tx lines using the standard RS232 USART protocol. It won't work because it can't toggle specific pins of serial port.

A similar programming interface (PonyProg with SIProg hardware) doesn't work with such adapters.

Yes, it is possible to program 8051 using an Arduino: Using Arduino to program at89s51