Can I use a roostick, which contains FTDI chip to program my arduino chip?
According to the product page/schematic, there is no 15.6V generated on the Roostick itself. It would normally come from the Roomba battery, but if you are using the roostick on it's own it won't be present.
So providing you have nothing else capable of providing >5V actually attached to the Roostick when you use it with the Arduino you won't fry your chip.

The FT232 is set to 3.3V logic level output, but you may not need any level conversion if your Arduino is running from 5V. I think the FT232 can handle inputs up to Vcc (which is at 5V from the USB) and the 3.3V high level should be fine for your Arduino input.
You can play safe in case the FT232R doesn't like an input voltage higher than it's VCCIO setting by putting a resistive voltage divider (to divide the 5V to 3.3V) on the line.
I'd just try it and see, if it doesn't work too well there are plenty of level shift ICs out there (and google knows plenty about simple discrete transistor type solutions too)  

Of course if your Arduino runs from 3.3V(?) this is irrelevant.
Check the FT232R datasheet for more details I only glanced briefly.  

No power connection is necessary since both the Roostick and Arduino have their own power source. 
So, you would just connect the Roostick/Arduino grounds together, and the serial lines: The pads on the Roostick are marked:R_TX, R_RX, and GND