It is generally better to use Linux dosbox rather than Windows dosbox under wine, unless it's a game like Eye of the Beholder, where Windows-specific extensions to dosbox are used (and even then, I just live without the extensions). Your distro probably has an adequate version of dosbox for this game.
Generic DOSBOX installation:
1) Extract the game. I usually use the Windows installer under wine for consistency, but you can also just use
innoextract. Note that even if your distro supplies innoextract, it may not be recent enough to support gog's version of Inno Setup. Extract with "innoextract -g -e setup_....exe" and the game will be in app/.
2) Delete the Windows dosbox: rm -rf DOSBOX
3) Create workable dosbox.conf files out of the gog-supplied files. Generally there is a dosbox<game>.conf and a dosbox<game>_single.conf. For network games, like Syndicate Plus, there may also be a *client.conf and *server.conf. There may be others for extra support, like *setup.conf. The dosbox<game>.conf contains most of the dosbox configuration. I normally use my own tuned dosbox config and just "diff -wub" with the supplied config to determine if there are any important changes; in that case, I copy those changes (and only those changes) into all of my game-specific configs. The main issue you need to deal with with the game-specific (single/client/server/etc) configs is that it's run relative to the DOSBOX dir in Windows, so convert ".." to "."; if CD data is used it might also be necessary to edit the CD descriptor files and change the mount command a bit (since it's case-sensitive, for one). As I said, I also tack on any relevant config changes from the main one.
In short, replace .. with . in *client.conf and *server.conf, and just use dosbox's default config for the rest.
4) Run dosbox using the created config file(s). If you want to use your own config, as I do, run "dosbox -userconf -conf <gameconfig>.conf". Here, gameconfig is your translation of the *client.conf or *server.conf file. If you want to use an edited version of the supplied base config file, just replace -userconf with "-conf dosbox<game>.conf" or whatever. Note that you can create/overwrite the default config by starting dosbox and using "config -wc" or "config -wc file". The former is unsafe if you don't want to accidentally overwrite your main config, though.
I personally do a lot more than what I described here, but only insane idiots would want that. For example, I use unionfs to allow me to keep the game data read-only and pristine, and I use wrapper scripts to simplify everything for me. I also enable munt where possible, and have my dosbox compiled for glide support (e.g. Redguards or Tomb Raider). Neither of these appear to be needed for Syndicate Plus, though.
I have never used networking in dosbox, so I can't help there if it doesn't work out of the box for you.
Edit: while I had no intention to provide info on how to make a desktop file/icon-based launcher, I should at least mention that you can convert the icon file (usually either gfw_high.ico or goggame-*.ico) using imagemagick's convert; e.g. "convert goggame-1207658992.ico goggame-1207658992.png". You will end up with more than one png file; pick the best one using an image viewer or whatever (my install script automatically picks the 256x256-RGBA one using "file").
Edit 2: In my attempt to finally update all of my Windows games, I noticed that games with cloud save support also mount another directory that obviously isn't necessary for Linux. I didn't even realize that dosbox had built-in overlay support, but I will continue to use overlayfs for myself.
