Don't try it as you will probably damage the Nano. A standard RS232 serial port could be outputting upto +15V to -15V and your Nano only wants TTL level of 0V to 3.3V/5V. You can buy cheap FTDI converters on eBay.
Its based on the MAX3232 so works at 5/3/3
Its much cheaper than the $9.95 ones in your links, is there a difference in terms of performance? Same IC in both I think.
Yes.
"Item will be shipped out within 48 hours upon receipt of your payment. Usually it will take 12-24 days on delivery."
Are you very patient?
Nano works at 5V.
Could also buy MAX3232 for 56 cents, add the 5 caps, and a connector, and be up & running in just a couple of days. http://www.dipmicro.com/store/MAX3232