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.
An engine should be usable in all four nacelles, the nacelles themselves cannot be swapped as they provide the reverser. The answer to this question seems to confirm this possibility.
Thrust reversal on this aircraft type consists in opening the bypass duct sideways, blocking 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 (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 XWB (similar to 900).
Based on RR picture
The cold flow duct (fan duct) is similar in both versions of the nacelles. The duct is fenced by two semi-annular walls behind the fan frame, where the flow is rushed by the fan. This is called a double D-duct (in reference to the shape of the semi-duct).
The two D-ducts of an inner nacelle contain the reverser panes and their associated components. This version is named the thrust reverser unit (TRU). The other version without the reverser is the fixed fan duct (FFD), it doesn't slide.
The TRU is visible behind the fan section, source
On the image above, the two halves of the TRU are open. Note the outer wall of the duct has some thickness, contrary to the FFD wall (see next). This volume is used to host the mechanism needed to move the panels rerouting the flow.
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.
More easy to see on this CG view:
Trent 900 duct and reverser. Based on TurboSquid models.
Top: 3D model by BlueGreen. Bottom: Another excellent 3D model by BlueGreen.
Left D-duct highlighted in blue.
Nacelle duct for cold flow. Based on image by Benny Zheng on Airliners.net