The reverser is not installed on the outer engines. As the linked question mentions: This allows to lighten the engines, this also prevents any unwanted and hazardous opening in flight as well as the costly check time for an unused feature.
The reverser used on the A380 is a cold flow type. It is merely part of the engine by itself. It is actually integrated in the nacelle structure channeling the bypass cold flow.
Thrust reversal on this aircraft type consists in opening the bypass duct sideways, interrupting the bypass flow in the duct and redirecting it through the opening. The opening is created by sliding the aft section of the nacelle case to expose a set of vanes. The blocked flow now escapes sideways and is guided forward by the vanes:
Principle of cold flow thrust reverser (source)
Thrust reverser on the A380 (source)
Engine, reverser, nacelle
The A380 is equipped with Trent 900 from Rolls-Royce or GP7200 from Engine Alliance. The thrust reversers are made by a different provider, Airbus-Aircelle (formerly Hurel-Hispano, now part of Safran), which also manufactures the nacelles. This is because the thrust reverser is really integrated in the nacelle rather than linked to the engine.
The reverser is in the cold / bypass flow behind the fan section which is the engine section with the largest diameter (in the Trent 900, about 3m!).
Hot flow, cold flow and reversed flow in a Trend 1000 (similar to 900).
Based on an excellent 3D model by Gandoza, on TurboSquid.
The cold flow duct is similar in both versions of the nacelles. The duct is fenced by two semi-annular walls, where the flow is rushed by the fan and the LP compressor.
The reverser features are just added or not at the end of the duct, just behind the fan frame.
Not the A380, but similar: Front panels can raise like petals and block the flow when the sleeve move aft (source)
Cold flow duct with reverser
The inner wall prevents all the flow to reach the core and the components and pipes around it, in order to not create damage or drag and heat. However a limited quantity of air is still directed toward the core for cooling.
Trent 900 duct and reverser.
Based on TurboSquid models.
Top image from the excellent 3D model by BlueGreen.
Bottom images from another excellent 3D model by BlueGreen.
Cold flow duct without reverser
Left half of the duct highlighted in blue.
Nacelle duct for cold flow. Based on image by Benny Zheng on Airliners.net