I am not a Ryanair pilot, but I will try to answer your question(s)
The Boeing 737-800 uses an inertial navigation system to fix it's position. This is basically a laser-guided gyro which measures all of the aircraft's movements from the moment it starts pushing back from the gate. Provided the IRS (Inertial Reference System) knows where it is starting from, it can keep track of the aircraft's position with an amazing degree of accuracy. When the system is started the crew enter the current INS position which is normally the a/c's parking stand. During flight the IRS also takes information from ground based stations such as VOR/DME and I believe it takes some data from GPS?
With regard to flying over London City, it is likely that the crew were following a Standard Arrival Route which is an extension to the airway system based on VOR/DME and ADF. During the approach phase ATC sometimes issue radar vectors which are headings that the crew have to steer, so it's possible that your arrival may have been somewhat random. However, the LOREL 2C and LOREL 2Q arrivals into Stansted would have brought you pretty close to London.
Hope this helps
