Manually.

The bonus campaign and developer commentary are the only things new to the extended edition, the other points are irrelevant and have been features of HK since it was released.

Yes, clicking the icon of the gun in the top left side, but this is in ShadowRun Hong Kong since the beginning, isn't new to the Extended Edition.

Shadowrun: Hong Kong - Extended Edition

Changelog for patch 3.1.1 / GOG-9 (Windows and Linux) / GOG-3 (Mac) (added 19 February 2016):

New Features, Improvements:
- Added new VFX to differentiate some spells and abilities.
- Minor animation, UI, VFX polish and improvements.

Core/Engine Bug Fixes:
- Resolved edge case issue that could interfere with the PDA from working as expected.
- Resolved edge case issue that could prevent levels, including Prosperity Tower, from successfully loading.
- Resolved issue that caused Drones to be able to be Detained.
- Resolved issue that caused some spells to show incorrect VFX when spell bouncing.
- Resolved issue that could cause actors to crouch mid move.
- Resolved issue that could cause game stall when performing a move and attack with Koshei.

Shadowrun Hong Kong Bug Fixes:
- Resolved edge case issue that could cause important NPC to forget about you under specific conditions in Heoi.
- Resolved issue that caused Matrix VFX to be seen from Meat Space of the Ronin mission.
- Resolved minor visual issues.

Shadows of Hong Kong Bug Fixes:
- Dialogue edit pass on Intro and Impound Hub first visit.
- Resolved edge case issue that could cause optional objective to fail incorrectly in Tai Po mission.
- Resolved edge case issue that could cause Party members to be placed incorrectly when moving from one area of the map to another in the Namazu mission.
- Resolved edge case issue that could prevent hostiles in final fight in the Tai Po mission from joining in the fight.
- Resolved edge case issue that could prevent the Detention mission from being completed.
- Resolved edge case issue that prevented important end game character from joining in the fight in Tai Po mission.
- Resolved issue that caused Achievement “Street Justice” to unlock prematurely.
- Resolved issue that caused Audio Commentary in Exit Stage Left mission to fail to play.
- Resolved issue that caused players to sit incorrectly in the Epilogue in some endings.
- Resolved issue that caused some hostiles to be unable to use their equipped weapons.
- Resolved issue that caused some hostiles to hold the incorrect weapon.
- Resolved issue that caused some mission items to persist after their mission has been completed.
- Resolved issue that caused Spider Tanks to become confused.
- Resolved issue that could allow Players to get stuck behind AI-only door in Namazu mission.
- Resolved issue that could cause hostiles to not enter the fight in the Tiger's Den mission.
- Resolved issue that could cause incorrect actors to be present in some end game conversations.
- Resolved issue that could cause some crew members to have incorrect dialogue options due to story choices.
- Resolved issue that could prevent Namazu mission from being completed when specific conditions were met.
- Resolved issue that could prevent optional objective to be unable to be completed in “Namazu” mission.
- Resolved issue that could show incorrect loading screen when returning from the Detention mission.
- Resolved issue that discouraged important end game character from healing in combat of Tai Po mission.
- Resolved issue that prevented a door from closing when expected in the Tiger’s Den mission.
- Resolved issue that prevented camera from zooming correctly when viewing boat in the Impound Lot.
- Resolved issue that prevented Police Data Analyst from moving as expected in the Tiger’s Den mission.
- Resolved issue that prevented room from being lit on fire in Detention mission.
- Resolved issue that prevented selling of Namazu paydata.
- Resolved issue that prevented some crew and hub actor conversations from being accessed.
- Resolved issue where sync of Spider Tank would not be interrupted upon actor death.
- Resolved minor visual issues.
- Resolved various other cosmetic text issues, such as typos and grammar errors.

The patched game works fine for me, though I notice a fairly heavy load (~40%) on 4 out of 8 processor cores. Didn't check the load before patching, though.

Shadowrun: Hong Kong - Extendend Edition

Changelog for patch 3.1.2 / GOG-10 (Windows and Linux) / GOG-4 (Mac) (added 04 March 2016):

Core/Engine Bug Fixes:
- Resolved animation issue that caused old dog models not to animate.
- Resolved issue that caused weapons to be reloaded upon loading save files.
- Resolved issue that could cause items with tag/variable prerequisite to be duplicated in equip screen.
- Resolved issue that prevented some locked items from being observed in the equip screen.

Shadowrun Hong Kong Bug Fixes:
- Resolved an edge case issue that could cause a conversation to end unexpectedly in the Showdown mission.
- Resolved issue that prevented some actors from equipping items as expected in the Showdown mission.
- Resolved various spelling, and other grammatical issues.

Shadows of Hong Kong Bug Fixes:
- Resolved issue that caused some actors to have their equipment duplicated on save/load in the Rude Awakening mission.
- Resolved issue that incorrectly displayed some player choices in save import screen.
- Resolved many grammatical and continuity issues, see notes below for more information.

Shadowrun Returns

Changelog for update 1.2.7(A) (added 22 March 2017):

- Added support for Cloud Saves in GOG Galaxy 1.2 and newer

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Shadowrun: Hong Kong - Extendend Edition

Changelog for update 3.1.2(A) (added 22 March 2017):

- Added support for Cloud Saves in GOG Galaxy 1.2 and newer

The Shadowrun Returns galaxy update did break the installer for the dragonfall DLC.
setup_shadowrun_dragonfall_dlc_2.3.0

I have described a possible fix for that here.
However it does require a registry edit. And not everyone can do that safely.

The standalone Windows installer for Shadowrun: Hong Kong - Extended Edition is listed on the website's download page as version 3.1.2.

However, upon downloading the files you will notice that the filename states 3.1.1 (9420).

This concerned me because v3.1.1 is also what is reported in the 'Add/Remove Programs / Uninstall or change a program' section of the control panel, and the 3.1.1 -> 3.1.2 patch didn't work.

The good news is that after installing the game, I went into the debug console and found the version number, which is actually 3.1.2. Despite these minor issues, it is the correct version of the game.

I didn't see this discussed elsewhere, so I wanted to make others aware of the error in hopes that this helps someone else avoid confusion.