Self-addressed and posted from Vatican City's post office outlet before exiting Vatican Museum. The staff was not really friendly, though.

The postmark was 11.10.2014 which I am quite sure that the staff must have forgotten to change the date slug on his franking device and the date of postage was 13 October 2014. But it made me quite happy because it was my birthday, all the reason I went to Rome for.
 
Cost to post from Vatican City to countries within Europe as of 2014 is €4.80 - you will need to fill a form which you have to ask from the counter. As not to upset the grumpy clerk, I let him chose the stamps. Three random stamps were put together, €4.00 + €0.70 + €0.10

The letter arrived on 20 October 2014 without any notification - must have been overlooked as the cover looks like a normal letter, nothing looks strange.



Note: For Vatican City's General Post Office, it is situated in the building on the right side of St. Peter's Basilica (on the left side if one facing the Basilica from St. Peter's Square). No entrance fee required and the staff are even polite and helpful than the small outlet by the exit from Vatican Museum.