Is there a way to reuse an old robot using arduino?
As to your question can I program in ArduinIDE - Yes. 


The main chip is from AVR ATMEGA16, 
the LCD is a 16x2 (pinout for use with the LCD library is enclosed) 
motor driver is (Big thanks to Juraj not a mystery) a ULN2803. Example to work with can be found here.


The circuit diagram enclosed:



How to use in ArduinoIDE:


First install the driver for prolific and makes sure your OS finds the robot  (done)


Then install the MightyCore which supports ATMega16 - there are detailed instructions how to do it
Start with a blink led example (pins from the schematic) compile and upload to test
Then a simple LCD example and so on.



Another posibility would be tor scrap the bot and salvage most parts.

If its feasable the Manual will tell you which parts can be removed without soldering
To get started you need the USB drivers for OS and the original SW to test wether the QBot is still ok from a hardware perspective,
Than you can salvage the parts from it to use with Arduino:
 
Than you have to make a new base plate (google for pics of 2-wheel robot base plates) and get a motor driver (I recommend for a beginner a cheap motor shield like an Adafruit V2 motor shield clone 5-7$). 
As you probably have downloaded ArduinoIDE latest version for your OS, get some example programs and start to play around.